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Zeta functions: from prime numbers to K-theory

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Zeta functions: from prime numbers to K-theory

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Content ZETA FUNCTIONS: FROM PRIME NUMBERS TO K-THEORY by Zhanhu Feng A Thesis Presented to the FACULTY OF THE USC DORNSIFE SCHOOL OF LETTERS, ARTS AND SCIENCES UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS (MATHEMATICS) May 2022 Copyright 2022 Zhanhu Feng Dedication I dedicate this thesis to my beloved parents, whom I am so lucky to have known my entire life. ii Acknowledgments My thesis supervisor, Professor Eric M. Friedlander, has provided me with invaluable support, instruction, and encouragement throughout the process. If it were not for his guidance and assistance, I could not have started to learn algebraic K-theory or completed my current thesis about its relationship with the Riemann zeta function. Professor Friedlander has made a significant difference in my academic journey at USC and my life by inspiring my interest in many algebra and topology fields, and by carefully reviewing my past papers. Professor Friedlander has been very kind to me. During my first visit to his office, he carefully took a book entitled “Zeta Functions: An Introduction to Algebraic Geometry” from his bookshelf and lent it to me, which I have included in the bibliography of my thesis. When I emailed him about my thesis updates, he would carefully review my drafts and provide many insightful suggestions and encouragement. He taught me to focus on a particular math topic while understanding related motivations in my future Ph.D. studies, which I have always been trying to do in writing my thesis. Furthermore, I also learned a great deal from Professor Friedlander’s papers in algebraic K-theory and other related areas. As well, I would like to express my deep gratitude to my master thesis committee, including Prof. Eric Friedlander, Prof. Susan Montgomery, and Prof. Sheldon Kamienny, who provided invaluable assistance in reviewing and revising my thesis. iii TableofContents Dedication ii Acknowledgments iii ListofTables vi ListofFigures vii Abstract viii Chapter1: Introduction: ProductFormulasandPrimes 1 1.1 Introducing the Riemann zeta functionζ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Sieve of Eratosthenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Prime-counting function and the infinity of primes . . . . . . . . . . . . . . . . . . 3 1.2 Prime zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Integral forms of the product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter2: AnalyticContinuationI:TopologicalExtension 13 2.1 A bigger picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Topological extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Application: Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 Application: Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Two integral representations for zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Corresponding results forζ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Chapter3: AnalyticContinuationII:FunctionalEquations 29 3.1 Dirichlet eta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Functional equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 Hurwitz’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Riemann’s functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.3 Hurwitz’s functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 An elementary proof of Riemann’s functional equation . . . . . . . . . . . . . . . . . . . . 42 3.3.1 Jacobi theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.2 Relation betweenΓ( s) andζ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.3 FromΦ( s) to Riemann’s functional equation . . . . . . . . . . . . . . . . . . . . . 46 iv Chapter4: ValuesofZetaFunctionsatIntegers 48 4.1 Basel problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 From Bernoulli numbers to Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Values of zeta functions at negative integers . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Values of zeta functions at nonnegative integers . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter5: Zerosofζ (s) 60 5.1 Zero-free regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2 Symmetry of zeros ofζ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Values ofζ (s) on the liness=0 ands=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.4 Miscellaneous modern results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4.1 Bohr-Landau theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4.2 Hardy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4.3 Approaching the critical line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Chapter6: AlgebraicK-theoryandZetaFunctions 77 6.1 Construction of higher algebraic K-theory groups K n . . . . . . . . . . . . . . . . . . . . . 77 6.1.1 An intuitive construction of then-th homotopy group . . . . . . . . . . . . . . . . 78 6.1.2 Classifying spaces and plus construction . . . . . . . . . . . . . . . . . . . . . . . . 82 6.2 Restricting to ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.3 Algebra prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.3.1 Global fields and Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.3.2 Ring of integers and localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.3.3 Tate twist and Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.4 Étale cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4.1 From singular homology to étale cohomology . . . . . . . . . . . . . . . . . . . . . 93 6.4.2 Local zeta function and Weil conjectures . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5 ClassifyingK n (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.5.1 K n (Z) for oddn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.5.2 Dedekind zeta function andK n (Z) for evenn . . . . . . . . . . . . . . . . . . . . . 104 6.5.3 Final conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.6 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Chapter7: Conclusions 112 Bibliography 115 Appendices 119 ChapterA: ACoherentIntroductiontoLowerK-theory 120 A.1 The Grothendieck group K 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.1.1 Group completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.1.2 Grothendieck group of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.2 The Whitehead group K 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.3 The Steinberg group K 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 ChapterB: Notes 136 v ListofTables 4.1 Bernoulli numbers and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 ζ (s) at positive odd integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.1 Gram’s first 15 zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.1 K-groups ofZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 vi ListofFigures 2.1 The loopC around the negative real axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 The “dog bone” contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 T(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 Zero-free region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Refined zero-free region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 vii Abstract In this thesis, the Riemann zeta function is introduced first through the sieve of Eratosthenes and product formulas, by which its relationship with prime numbers is illustrated. Following this, the analytic contin- uation of the Riemann zeta function, as well as the Hurwitz zeta function, is discussed from two different perspectives: contour integration and functional equations. Based on these foundations and the construc- tion of similar zeta functions, numerous results, such as the Basel problem, Bernoulli polynomials, and Apéry’s constant, are presented about special values of the Riemann zeta function, including the distribu- tion of its zeros. Lastly, some concepts of algebraic K-theory and étale cohomology are outlined, whereby a connection between the Riemann zeta function and K-groups of rings of integers is demonstrated. More- over, the Riemann zeta function is further generalized to the local zeta function and the Dedekind zeta function in the classification of K-groups K n (Z) of the ring of integers. A brief introduction to lower K-groups can be found in Appendices, which serves as a prerequisite to understanding the chapter about K-theory. Also included in this thesis is an original concept called the topological extension model that connects many seemingly unrelated concepts about zeta functions and K-theory. viii Chapter1 Introduction: ProductFormulasandPrimes 1.1 IntroducingtheRiemannzetafunctionζ (s) As we know in elementary mathematics, aprime (number)p is defined as a natural number greater than 1 that cannot be factored into a product of two smaller natural numbers. Denote the set of all prime numbers asP. In this section, we will take a close look at the distribution of primes whence we introduce the Riemann zeta function, which is the main topic of this thesis. 1.1.1 SieveofEratosthenes First, we may use an ancient method, called thesieveofEratosthenes, to easily screen primes as follows: StepI: Write down all the integers from2 (including2; here we write these numbers until50 as an example). 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Step II: Starting with2, keep2 and remove all multiples of2. 1 2 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 StepIII: Starting with3, namely the next number untouched after2, keep3 and remove all multiples of3. 2 3 5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 StepIV : Starting with5, namely the next number untouched after3, keep5 and remove all multiples of5. 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 49 ... Step k: Starting withp, namely the next number untouched after the one that is the least common factor of the number removed in Step k-1, keepp and remove all multiples ofp. 2 ... If we keep doing this process till the last number retained, we will filter out all the prime numbers from 1 to50: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 And if we further continue this process forever, we’ll be able to filer out all primes among integers. 1.1.2 Prime-countingfunctionandtheinfinityofprimes To further research the distribution of primes, we introduce a new function. Definition1.1.1 (Prime-counting function). Define π (x) :=|{p∈P :11. By assumption,s cannot be prime, and thus it is composite. 3 However, by definition any prime p dividings will have remainder 1, sos is not composite, a contra- diction. Therefore, we have lim x→+∞ π (x)=+∞. In 1737, motivated by the above sieve of Eratosthenes, Euler [13] provided another proof of this limit as follows. Let’s first consider the p-series: p(s):= ∞ X n=1 1 n s ≡ 1+ 1 2 s + 1 3 s +··· (1.1) We find 1 2 s p(s)= ∞ X n=1 1 (2n) s ≡ 1 2 s + 1 4 s + 1 6 s +··· ( 1 2 s (1.1)) (1.1)− ( 1 2 s (1.1)) ⇒ 1− 1 2 s p(s)=1+ 1 3 s + 1 5 s +··· (1.2) Likewise, 1 3 s 1− 1 2 s p(s)= 1 3 s + 1 9 s + 1 15 s +··· ( 1 3 s (1.2)) (1.2)− ( 1 3 s (1.2)) ⇒ 1− 1 3 s 1− 1 2 s p(s)=1+ 1 5 s + 1 7 s +··· (1.3) ... Repeating the process, we conclude 4 (1.m)− 1 n s (1.m) ⇒ ··· 1− 1 5 s 1− 1 3 s 1− 1 2 s p(s)= X n̸=2k,n̸=3k n̸=5k,... 1 n s ≡ 1. (1.(m+1)) In other words, we have deduced ∞ X n=1 1 n s = Y p 1− p − s − 1 . (1.4) This equation is known as Euler’s product formula. Here Q p is short for Q p∈P . This formula unexpectedly establishes a relationship between thep-series and all primes – it is unusual for a series over natural numbers to be connected with the distribution of primes. As a result, the series on the left-hand side has a great potential to be further examined, and that’s what we will do in the remaining part of the thesis. Let’s give it a special name: thezetafunction, denotedζ (s). Notice by the current definition, we haveζ (s)=∞, namely this series diverging, whens<1. 1.2 Primezetafunction Before deeply exploring the above ζ (s), first let’s take advantage of this Euler’s product formula we de- duced and see what kind of results we can further obtain. Taking the natural logarithm, we get lnζ (s)= X p ln 1− p − s − 1 ,ℜs>1. (1.5) Hereℜs>1 is to guarantee the convergence of the series on the right-hand side. Expanding by Taylor’s formula: ln 1 1− x = ∞ X n=1 x n n ,− 1<|x|<1, (1.6) 5 we have lnζ (s)= X p ∞ X n=1 1 np sn = X p ( 1 p s + 1 2p 2s + 1 3p 3s +··· ). (1.7) We find the first term P p 1 p s resembles the zeta functionζ (s) except only the value ofn is narrowed from all positive integers to all primes, so let’s also give it a name in the following definition: Definition1.2.1 (Prime zeta function). We call the functionP (s) := P p 1 p s = 1 2 s + 1 3 s + 1 5 s +··· the primezetafunction, which was originally proposed by Glaisher in 1891 ([19]). By (1.7), lnζ (s)= X p 1 p s + X p ∞ X n=2 1 np sn =P (s)+ ∞ X n=2 1 n X p 1 p sn . (1.8) The second equal sign holds since P ∞ n=2 1 np sn is uniformly convergent (and it is easy to prove its absolute convergence for eachs by the ratio test). Note that we must have 1 p s <1 wheneverℜs≡ σ > 1 to let (1.8) make sense, but this is trivial since 1 p s = 1 |p σ +it | = 1 p σ <1∀σ > 1 always holds. And from this we get X p 1 p sn ⩽ X p 1 |p (σ +it)n | = X p 1 p σn ⩽ X p 1 p n ⩽ ∞ X k=2 1 k n . Then we have ∞ X k=2 1 k n < ∞ X k=2 Z k k− 1 1 t n dt= Z +∞ 1 1 t n dt= 1 n− 1 . (1.9) Hence X p 1 p sn < 1 n− 1 . 6 Therefore in (1.8), 0< ∞ X n=2 1 n X p 1 p sn ⩽ ∞ X n=2 1 n X p 1 p sn < ∞ X n=2 1 n 1 n− 1 = ∞ X n=2 1 n− 1 − 1 n =1. Notice by definition, ζ (s) = 1 + 1 2 s +··· > 1 so lnζ (s) > 0. Likewise, the previous double sum P ∞ n=2 1 n P p 1 p sn >0. Hence by (1.8) we conclude P(s)<lnζ (s) 0 is a subseries of ζ (2) ∈ (0,∞), which is another way to showP(2)<∞. More generally, sinceζ (s) all converge for anyℜs>1, likewise we have: Property1.2.4. P (s) converges for allℜs>1. By this property, we further obtain: Theorem1.2.5. X p p − 1 ∼ ln X n n − 1 ! ∼ lnln(∞). Proof. Lettings=:1 in (1.8), we have ln X n n − 1 ! = X p p − 1 + X p 1 2p 2 + 1 3p 3 +··· ÷ both sides by P p p − 1 = ============= ⇒ ln P n n − 1 P p p − 1 =1+ P p 1 2p 2 + 1 3p 3 +··· ( Prop1.2.4 < ∞) P p p − 1 ( Prop1.2.2 = ∞) def ⇔ X p p − 1 ∼ ln X n n − 1 ! . By Taylor’s formula (1.6), ln X n n − 1 ! ∼ lnln(∞). This theorem shows P p p − 1 diverges in the same waylnln(∞) diverges. 8 Likewise, we easily show the partial sum has a similar result: P p⩽n 1 p ∼ lnln(n); equivalently, P p⩽n 1 p =ln(ln(n))+O(1). (See [18].) 1.3 Integralformsoftheproductformula Going back to the Euler product formula (1.4), now instead of applying Taylor’s formulas as we did above, we may also utilize the prime-counting function to get: lnζ (s)= X p ln 1− p − s − 1 = ∞ X n=2 (π (n)− π (n− 1))ln 1 1− n − s = ∞ X n=2 π (n)ln 1 1− n − s − ∞ X n=2 π (n)ln 1 1− (n+1) − s = ∞ X n=2 π (n) − ln 1− n − s +ln 1− (n+1) − s = ∞ X n=2 π (n)ln 1− x − s x=n+1 x=n = ∞ X n=2 π (n) Z n+1 n s x(x s − 1) dx. By definition, we may easily verify a property of π (x): π (n) Z n+1 n f(x)dx= Z n+1 n π (x)f(x)dx. Hence ∞ X n=2 π (n) Z n+1 n s x(x s − 1) dx= ∞ X n=2 Z n+1 n π (x)s x(x s − 1) dx. Together, we conclude lnζ (s)= Z +∞ 2 sπ (x) x(x s − 1) dx. This establishes a relationship between the zeta function and the prime-counting function, which in a way further strengthens the relation ofζ (s) with primes. 9 In the 1859 [37] paper, Riemann introduced another prime-counting function defined as a step function J(x)(x⩾0) such that                      J(0):=0, J(x) increases by 1 n once its variablex passes throughp n (wherep∈P) asx increases, J(x):= 1 2 [J(x − )+J(x + )] ifx is a discontinous point. In other words, J(x):= X n⩾1 X p n ⩽x 1 n . Also by definition of the prime-counting function, J(x)= ∞ X n=1 1 n π n √ x ≡ π (x)+ 1 2 π ( √ x)+ 1 3 π ( 3 √ x)+ 1 4 π ( 4 √ x)+ 1 5 π ( 5 √ x)+··· Notice this is just a finite sum for a given x. One main result of Riemann’s 1859 paper is this formula: lnζ (s)=s Z ∞ 0 J(x)x − s− 1 dx. Proof. In Equation (1.9) above, we have applied a basic calculus fact: s Z ∞ n x − s− 1 dx=− x − s | ∞ n = 1 n s . Now by this equation, we get, for any sequence{a n } ∞ n=0 , X n a n n s =s X n a n Z ∞ n x − s− 1 dx. (1.11) 10 Again, observe by basic calculus that Z ∞ n x − s− 1 dx= Z ∞ 1 χ (n,x)x − s− 1 dx whereχ (n,x):=            1 ifn⩽x 0 ifn>x is a specially-defined step function. Indeed, this follows from Z ∞ 1 χ (n,x)x − s− 1 dx=( Z n 1 + Z ∞ n )χ (n,x)x − s− 1 dx = Z ∞ n χ (n,x)x − s− 1 dx= Z ∞ n x − s− 1 dx sinceχ (n,x)=0 ifx∈(1,n). Hence, continuing from (1.11), X n a n n s =s X n a n Z ∞ 1 χ (n,x)x − s− 1 dx =s Z ∞ 1 X n a n χ (n,x)x − s− 1 dx =s Z ∞ 1 X n⩽x a n x − s− 1 dx (1.12) by definition of χ (n,x). Note that by the Tonelli theorem (Theorem 2.37 in [15]), we can interchange the summation and integral above. By (1.7), lnζ (s)= X p ∞ X n=1 1 np sn = X k⩾1 X p 1 kp sk . (1.13) 11 Hence let’s takea n =:            1 k for alln=p k 0 else to get a close form of the right-hand side of Equation (1.13) from Equation (1.12) : X ′ n 1 kn s ≡ X p 1 kp sk =s Z ∞ 1 X p k ⩽x 1 k x − s− 1 dx where the first X ′ n meansn only takes values of the formp k . Accordingly, (1.13) becomes lnζ (s)= X k⩾1 s Z ∞ 1 X p k ⩽x 1 k x − s− 1 dx =s Z ∞ 1 X k⩾1 X p k ⩽x 1 k x − s− 1 dx def = s Z ∞ 1 J(x)x − s− 1 dx. Notice by definition J(x)≡ 0∀0⩽x⩽1, so we conclude lnζ (s)=s Z ∞ 0 J(x)x − s− 1 dx. 12 Chapter2 AnalyticContinuationI:TopologicalExtension 2.1 Abiggerpicture Before presenting the powerful tool in complex analysis known as the analytic continuation, let’s intro- duce a model that allows us to see a big picture in math and deepen the understanding of the notion of analytic continuation. Typically speaking, this model serves as a compass to predict possible future di- rections of modern math as a succinct generalization of most modern math results. Besides the analytic continuation we are about to introduce in this chapter, we will also use this model many times later in the thesis motivating us to introduce new concepts and theorems. In basic modern algebra, we have defined the semigroup as a set having an associative operation. Moreover, amonoid is a semigroupS that has a two-sided identity element1; i.e. 1s = s = s1∀s∈ S. Like in groups, a monoid homomorphism is naturally defined as f : S → S ′ such that the equations            f(st)=f(s)f(t)∀s,t∈S f(1)=1∈S ′ hold. In 1904, J.-A. de Séguier first used the term, semigroup (in French), in Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups), which was much later than the invention of the termgroup; the invention of groups was motivated by generalizing some “core” properties in basic numerical calculations. So we can see the definition of monoids above is 13 obtained by adding some restrictions on a semigroup, in order for it to be “more like a group”. Therefore, we can abstract an extension model from it. 2.1.1 Topologicalextension Definition2.1.1 (Topological extension). In general, atopologicalextension is the process of defining a new math conceptG from abstracting some chosen propertiesP 1 ,P 2 ,... (which we consider to be the “core” properties) of a given (known) conceptF such that the group of all such core properties is equivalent to the definition of F . We also callG a topological extension ofF . As a fundamental example, the construction of “topological space” is a topological extension. We found three “core” properties of open sets in any familiar metric space X: both X and∅ are open, any union of open sets is open, and any finite intersection of open sets is open. Notice under a metric space, if a set satisfies these three properties then this set must be open, which is what makes them the “core” properties as in Definition 2.1.1. Then we abstracted these three basic properties from open sets in F = “metric space” and innovated a more generalized notion, which we happened to also call “open sets”, inG= “topological space”. Note that the naming of “open sets in G” turned out to be magical because it automatically mo- tivated us to examine and invent the parallel concepts, such as “closed sets”, “neighborhoods”, “closure”, etc., in a topological space. The example above is what motivated us to call this kind of extension a “topological” extension in general; as we can see, the naming of “topological extension” referenced to that of “open sets in a topo- logical space”. Also from this example, we can see “equivalent” in the core properties is necessary here, because: if we defined open sets in a topological space X as sets that only satisfies the first condition “both X and∅ are open”, then this definition would not be identical with (indeed much broader than) open sets in the original metric space sense when setting X = “a metric space”. So it would be considered a bad, “degenerate (singular)” extension. Here are a few other parallel examples: 14 Example 2.1.2 (Odd and even functions). For power functions f(x) = x n (n ∈ Z), we naturally call those functions with n even (...,− 2,0,2,4,...) as even functions since the power of the variable is even, and those with n odd as odd functions. We find this function satisfies ∀x,f(− x) = f(x) when f(x) is even (i.e. n is an even number by the definition above), and satisfies f(− x) =− f(x) whenf(x) is odd (i.e., n is an odd number). Also in contrary, in the domain of all power functions, once a function f(x) = x n (n ∈ Z) satisfies f(− x) = f(x) then f(x) must be even (i.e., n must be even), and once it satisfies f(− x)=− f(x) thenf(x) must be odd (i.e.,n must be odd). Hence following Definition 2.1.1, we may regard these two properties above as the “core” properties of power functions, and define any function (not necessarily a power function) that satisfies the property f(− x)=f(x) as anevenfunction; likewise we define any function that satisfies the property f(− x)=− f(x) as anoddfunction. We see these two definitions of even (odd) functions are different but agree in the domain of power functions (for n∈Z), and the second definition is a topological extension of the first definition. Indeed, this example shows how the terms even/odd functions in modern math language were invented. 1 Example2.1.3 (Weak derivatives and weak solutions). (I) Supposef ∈C 1 ([a,b]) andφ is any infinitely differentiable function with φ(a)=φ(b)=0. By the integration by parts (IBP) formula, we have R b a f(t)φ ′ (t)dt=− R b a f ′ (t)φ(t)dt. Now we want to extend the restriction of functionf fromC 1 ([a,b]) to a much broader spaceL 1 ([a,b]), while still preserving the definition of derivatives. We are enabled to do that by the topological extension model. Observe this IBP equation above is a property off ′ (t); moreover, by the product rule (which is the motivation of the integration by parts formula), we see the derivativesf ′ (t) are the only functions inC 1 ([a,b]) satisfying this IBP equation. Hence we may consider this equation to be the “core property” of derivatives, and define the derivativef ′ (called the weak derivative) of a functionf inL 1 ([a,b]) to be any function satisfying this equation. 15 Further, from this motivation, we may extend our dimension from1 ton and extend the closed interval [a,b] to any nonempty open setΩ in a Euclidean space. Supposef ∈L 1 loc (Ω) andφ∈D(Ω):= C ∞ c (Ω) . We define the α -thweak(partial)derivative off to be the functionD α f ∈L 1 loc (Ω) such that Z Ω fD α φ=(− 1) |α | Z Ω D α fφ∀φ∈C ∞ c (Ω) whereα is a multi-index,|α | := P n i=1 α i , andD α φ := ∂ α 1 ∂x α 1 1 ··· ∂ α n ∂x α n n φ i.e. = ∂ |α | φ ∂x α 1 1 ...∂x α n n . Note in this case, φ has compact support inΩ and thus vanishes near the boundary∂Ω ; that is why we choseφ∈C ∞ c (Ω) . We may easily show a weak partial derivative is unique up to a set of measure zero by this definition; a proof can be found on Page 257 in [14]. Still further, if we extend our functions to generalized functions (distributions) T ∈ D ′ (Ω) (namely, T :D(Ω) →C is a continuous linear functional onD(Ω) ), we will get the following definition accordingly: theweakderivative (a.k.a.distributionalderivative) ofT is defined as the distribution D α T satisfying (D α T)(φ)=(− 1) |α | T (D α φ) ∀φ∈C ∞ c (Ω) . Sometimes we are accustomed to writingD α T as∂ α T , and writing this equation above as ⟨∂ α T,φ⟩=(− 1) |α | ⟨T,∂ α φ⟩. They mean the same. Again, naturally we have, forf ∈C |α | (Ω) , (D α T f )(φ) def = (− 1) |α | Z Ω (D α φ)fdx IBP = Z Ω (D α f)φdx def = T D α f (φ)∀φ∈C ∞ c (Ω) whereT f (φ) := R Ω f(x)φ(x)dx in general; namely the notion of weak derivative is indeed a valid topo- logical extension which agrees with the notion of derivative in the classical sense. 16 (II) Similarly, given a ij ,b i ,c ∈ L ∞ (U)(i,j = 1,...,n) and f ∈ L 2 (U); suppose v ∈ C ∞ c (U) is a smooth test function andu∈C ∞ (U) is a smooth solution to the elliptic equation Lu:=− n X i,j=1 a ij (x)u x i x j + n X i=1 b i (x)u x i +c(x)u=f. (2.1) Now let’s multiply both sides of the equation byv and then integrate overU; again by IBP in the first item, we’ll obtain B[u,v]:= Z U n X i,j=1 a ij u x i v x j + n X i=1 b i u x i v+cuvdx= Z U fvdx=:(f,v). (2.2) (2.2) is deduced from (2.1) but we observe that (2.2) can still make sense and hold if we enlarge our domain from            u∈C ∞ (U) v∈C ∞ c (U) to allu,v ∈ H 1 0 (U) := W 1,2 0 (U) := “C ∞ c (U) inW 1,2 (U)”. Therefore, our topo- logical extension model motivates us to define the new u∈H 1 0 (U) to be aweaksolution of (2.1) if (2.2) holds for allv∈H 1 0 (U). Moreover, we easily verify the weak solution is also unique by the Lax-Milgram theorem (see Theorem 1 to Theorem 5 in Chapter 6 of [14]). Example2.1.4 (Oxidation and reduction reactions). The chemical reaction in which substance A is syn- thesized with oxygen is called theoxidationreaction, and its reverse process (the reaction in which oxide ions are lost from substance A and oxygen is evolved) is called thereductionreaction. One property of the oxidation reaction is that A loses electrons and thus its valence will increase. Hence, in the domain of all reactions (not necessarily with oxygen involved), we call any reaction about losing electrons as anox- idationreaction, by our topological extension model. Likewise, we call any reaction about the substance A gaining electrons as areductionreaction. Indeed, topological extensions can be found everywhere in any math branch. Derivations ofC ∞ p (M) defined by the product (Leibniz) rule (Definition 6.4.5 below), the definition of u m converging tou in the 17 Sobolev space W k,p (U), “the norm” and “orthonormal basis” in a Hilbert space, “alphabet” and “words” in free groups, “Cauchy sequences in a metric space”, etc., are all topological extensions. Of course, there are also other types of extension models; for example, we don’t consider the invention of the imaginary unit (extending our universeR by forcingi 2 =− 1) to be defined by topological extensions. And we will introduce another extension model called thetranslatedextension in Appendix A. From all these examples, we can clearly see that the topological extension model is the “root” of most abstractness in modern mathematics, which clearly tells us where these abstract math definitions come from and where they will go in the future. As we can see, this topological extension model provides explicit motivations for many math concepts in various branches. 2.1.2 Application: Regression Now let’s go even further and provide an important practical application of the topological extension model: PatternR(Regression). In modern mathematics, we usually make a group of definitions through starting from a direct generalization G of a familiar, intuitive concept F, then extending this new concept G in the broadest way that we can think of to B, and lastly gradually regressing B (the broadest concept) to G by addingmorerestrictionsstepbystep(e.g. viansteps). Wemayidentifytheprocessofaddingeachrestriction as a mapf; i.e.,G=f n (B). Note that these extensions inPatternR can either refer to topological extensions or any other exten- sions. In particular, in the generalization about monoids above, we haveG = “group”,B = “semigroup”, and f(B) = “monoid”. As we create more and more math definitions and properties, we can then try thinking backwards to find what a possible f − 1 (B) might be, namely to further enlarge or extendB. As a seemingly simple summary,PatternR has certainly been a wonderful way to invent and extend modern 18 math (e.g., “topological extension” creates a novel math branch while different groups of the following “regressions” enrich this branch), which will be proved (at least supported) by the following examples. Example 2.1.5 (Foundation of point set topology). Open sets in F =: R n can be defined intuitively as related to life, and can be topologically extended to G =: “metric space” (by abstracting the notion of “distance”) and further, as stated in the fundamental example above, to B = “topological space” (by abstracting the three properties of open sets); then we build f(B) = T 0 (Kolmogorov), f 2 (B) = T 1 (Fréchet), f 3 (B) = T 2 (Hausdorff), etc., by the separation axioms, in order to restrict B back to the familiarG = “metric space” step by step (T 0 ⊋ T 1 ⊋ T 2 ⊋ ...). Indeed, other topological constructions, such as connectedness and axiom of countability (say,g(B):= “first-countable space”), all follow the same pattern as we can easily verify: in a way, these parallel groups of notions together have formed the entire basis of the topology field. Example 2.1.6 (Foundation of modern algebra). In algebra, let F =: Q; then a possible way is G = “field”, B = “integral domain”. As in the three examples in the previous subsection, F → G here is a topological extension by abstracting some core properties ofQ, such as the inverse in multiplication, to get the definition of fields in general; so is G→B. Hence we have inventedf(B) = UFD,f 2 (B) = PID, f 3 (B)= “Euclidean domain”, etc., in an attempt to restore all the properties a field has in general. Indeed, the constructions of “group→ ring→ field” are also motivated by this pattern, and they together have formed the entire basis of the modern algebra field. Exercise. Think of what a possible f − 1 (B) could be in each of the examples above? And find more examples that followPatternR. 2 2.1.3 Application: Analyticcontinuation With the foundation of the previous two subsections, let’s get into our main topic as another application of the topological extension model. We have known the domain of convergence of functionζ (s)= P ∞ n=1 1 n s 19 is{s ∈ C : ℜs > 1}. Whenℜs ⩽ 1, ζ (s) = ∞ by our previous definition, so we now don’t consider ℜs⩽1 to be properly defined. Following the previous topological extension model, we choose F =: “the zeta functionζ (s)”,P =: “analyticity”, so we’ll get a broader definition G= “a function (we may still call a zeta functionζ (s) as we did for open sets, but now to highlight the difference let’s call it z(s) for now) that is analytic with some broader domain of definition”. In other words, literally this G is not the original F but just equivalent toF whenF andG coincides; that is, z(s):=            ζ (s) ifℜs>1, some function preserving analyticity (P ) inC ′ ifs∈C ′ −{ℜ s>1}. Here we denoteC ′ :=C−{ 1} 3 ; we will explain below why we especially take away the point1. In fact, finding this z(s) is exactly what Riemann did in his 1895 paper which set a foundation for modern analytic number theory. So we can feel how powerful this application of the topological extension model is. In analysis, we call this particular topological extensionanalyticcontinuation. But why do we especially care about this particular caseP =: “analyticity”? Why don’t we first discuss “continuous continuation” whenP =:“continuity”? Well, in the latter case, there will be infinitely many possible G since the only requirement is that G is continuous while it coincides with the original zeta function: that is, each well-defined (thus single-valued) continuous function defined in C that coincides withζ (s) in{ℜs>1} can serve as a candidate forG. Hence, this case is too trivial for mathematicians to research. On the other hand, what’s special about the analytic case is: Property2.1.7. Analytic continuations are unique. The proof is trivial by the famous unique theorem in complex analysis: 20 Proof. SupposeF 1 andF 2 are two analytic continuations off. IfU ⊃ V is a connected domain of both analytic functionsF 1 andF 2 such that∀z ∈ V ,F 1 (z)=F 2 (z)=f(z); then by the uniqueness theorem we haveF 1 =F 2 on all ofU. That is, analytic continuations are unique (up to function equality). As we have seen in the previous examples such as weak derivatives and weak solutions, the cases we care about usually possess certain uniqueness. If the uniqueness for a topological extension doesn’t exist, we usually categorize/integrate all cases for this notion and then define them as one set, so this notion will possess uniqueness up to this set. This is also a general trick in modern math. So, naturally, our next goal is to find this unique C-analytic functionz(s) equal toζ (s) for allℜs>1 (i.e., the original domain). One possible way to do that is to use the gamma function: Γ( s) def = Z ∞ 0 t s− 1 e − t dt∀ℜs>1 t=:nu ⇒ Γ( s)= Z ∞ 0 n s u s− 1 e − nu du. In order to construct the zeta function, we multiply both sides by 1 n s : Γ( s)· 1 n s = Z ∞ 0 u s− 1 e − nu du. Then taking the sum fromn=1 to∞, ∞ X n=1 Γ( s)· 1 n s = ∞ X n=1 Z ∞ 0 u s− 1 e − u n du= Z ∞ 0 u s− 1 ∞ X n=1 e − u n du. Note the integral absolutely converges so the summation and integration are exchangeable. Thus Γ( s)ζ (s)= Z ∞ 0 u s− 1 e − u 1− e − u du= Z ∞ 0 u s− 1 1 e u − 1 du 21 def ⇔ζ (s)= 1 Γ( s) Z ∞ 0 u s− 1 e u − 1 du∀ℜs>1. We know the gamma function is meromorphic inC with poles at0,− 1,− 2,... and without zeros. So the right-hand side of the last equation can serve asz(s). But note that whens=1,ζ (1)= 1 Γ(1) R ∞ 0 1 e u − 1 du= R ∞ 0 1 e x − 1 dx=[ln|e x − 1|− x] ∞ 0 =∞, so the newz(s) is actually defined in C ′ =C−{ 1}; namelyz(s) is still not entire since it’s undefined at one point s=1. As we did for open sets, henceforth, unless we emphasize the analytic continuation especially, we shall identify “F =ζ (s)” and “G=z(s)” (as described in the first paragraph for the topological extension model) in the rest of the thesis. 2.2 Twointegralrepresentationsforzetafunctions In this section, we are going to examine a more generalized function that includes the Riemann zeta func- tion as a special case. For simplicity, we writeℜs =: σ andℑs =: t (following Riemann’s notations in [37]). Definition 2.2.1 (Hurwitz zeta function). Define ζ (s,a) := P ∞ n=0 1 (n+a) s ,0 < a ⩽ 1 for σ > 1. We call this double-variable function defined by the infinite series the Hurwitzzetafunction. In particular, ζ (s,1)=ζ (s). Before we examine the basic properties of this new function, let’s introduce a lemma, quoted from Theorem 1 in Chapter 5 of [1]: Lemma2.2.2 (Weierstrass’s Theorem). Supposef n (z) is analytic in some regionΩ n and the sequence{f n (z)} converges tof(z) inΩ uniformly on everycompactsubsetofΩ . Thenf(z)isanalyticinΩ . Moreover,f ′ n (z)→f ′ (z)uniformlyoneverycompact subset ofΩ . 22 Property2.2.3. (I)ζ (s,a) converges absolutely inΩ:= {s∈C:σ > 1}. (II)ζ (s,a) converges uniformly in{s∈C : σ > 1+ϵ }∀ϵ > 0; moreover, it converges uniformly on every compact subset ofΩ . (III)ζ (s,a)isanalyticinΩ . Moreover,wehavethes-derivativeζ ′ n (s,a)ofζ n (s,a):= P n k=0 1 (k+a) s converges uniformly toζ ′ (s,a) on every compact subset ofΩ . Proof. (I) In general,∀z∈C,|z s |= e (σ +it)lnz = e σ lnz e itlnz = e σ lnz =|z σ |. Hence |ζ (s,a)|⩽ ∞ X n=0 1 (n+a) s = ∞ X n=0 1 (n+a) σ . Forσ > 1 (inΩ ), P ∞ n=0 1 (n+a) σ converges⇒|ζ (s,a)| converges⇔ζ (s,a) converges absolutely, (I) proved. (II) Forσ > 1+ϵ , P ∞ n=0 1 (n+a) σ ⩽ P ∞ n=0 1 (n+a) 1+ϵ which converges uniformly⇒ζ (s,a) converges uniformly. Hence the case for any compact subsets ofΩ follows, (II) proved. (III) The sequenceζ n (s,a) := P n k=0 1 (k+a) s ,0 < a⩽ 1 converges to the limit functionζ (s,a). Obvi- ously,ζ n (s,a) is analytic inΩ . Similar to (II), we easily knowζ n (s,a) also converges uniformly on every compact subset ofΩ . Hence by Lemma 2.2.2 (Weierstrass’s Theorem), (III) follows. As the extension process forζ (s) shown in the previous subsection §2.1.3, we have: Theorem2.2.4. Γ( s)ζ (s,a)= Z ∞ 0 x s− 1 e − ax 1− e − x dx∀σ > 1. (2.3) Proof. ∀σ > 1,Γ( s) def = Z ∞ 0 e − x x s− 1 dx x=:(n+a)u = (n+a) s Z ∞ 0 e − (n+a)u u s− 1 du ⇒ Γ( s) (n+a) s = Z ∞ 0 e − (n+a)u u s− 1 du 23 P ∞ n=0 ⇒ ζ (s,a)Γ( s)= ∞ X n=0 Z ∞ 0 e − (n+a)u u s− 1 du (1) = Z ∞ 0 u s− 1 e − au ( ∞ X n=0 e − nu )du (2) = Z ∞ 0 u s− 1 e − au ( 1 1− e − u )du= Z ∞ 0 e − ax x s− 1 1− e − x dx. Here in (1) above, the infinite sum and integral are interchangeable by the Tonelli theorem, and (2) is by the infinite geometric series formula. Equivalently, we have Theorem2.2.5. (I)∀σ > 1, ζ (s,a) = Γ(1 − s)I(s,a), where I(s,a) := 1 2πi R C z s− 1 g(z)dz and g(z) := e az 1− e z . (That is, I(s,a)= 1 2πi R C z s− 1 e az 1− e z dz.) HeretheintegrationcontourC isaninfinitepaththatstartsat ∞− ri,encircles z =c and returns to∞+ri forr⩾c, as shown in Figure 2.1. (II)I(s,a) is entire. Proof. (I) First, we divide the contourC into three parts as shown below (so the radius ofC 2 isc): Figure 2.1: The loopC around the negative real axis We have I(s,a)=: 1 2πi Z C z s− 1 g(z)dz whereg(z):= e az 1− e z 24 ⇒2πiI (s,a)=( Z C 1 + Z C 2 + Z C 3 )z s− 1 g(z)dz. (i) Z C 1 z s− 1 g(z)dz z=re − πi onC 1 ======= Z c −∞ (re − πi ) s− 1 g(− r)e − πi dr = Z c −∞ r s− 1 e − πis g(− r)dr →− Γ( s)ζ (s,a)e − πis asc→0, by Equation (2.3). (ii) Z C 2 z s− 1 g(z)dz z=ce θi onC 2 ======= (− π ⩽θ ⩽π ) Z π − π c s− 1 e iθ (s− 1) g ce iθ ce iθ idθ =c s− 1 i Z π − π e iθ (s− 1) (ce iθ )g ce iθ dθ →0 asc→0, since it is easy to verifyg(z) is analytic in the region|z|<2π except for a simple pole atz =0 and thus(ce iθ )g ce iθ <∞ asc→0. (iii) Z C 3 z s− 1 g(z)dz z=re πi onC 3 ======= Z ∞ c (re πi ) s− 1 g(− r)e πi dr = Z ∞ c r s− 1 e πis g(− r)dr →Γ( s)ζ (s,a)e πis asc→0, by (2.3) again. Therefore, 2πiI (s,a)=Γ( s)ζ (s,a)(e πis − e − πis ) where we find (e πis − e − πis )=cosπs +isinπs − cosπs +isinπs =2isinπs. So 2πiI (s,a)=Γ( s)ζ (s,a)2isinπs (1) = ζ (s,a) 2πi Γ(1 − s) ⇒ζ (s,a)=Γ(1 − s)I(s,a). 25 Here (1) above represents the following Euler’s reflection formula (see [1]), which we will discuss in Lemma 3.2.6: Γ( s)Γ(1 − s)= π sin πs . (II) First we again splitI(s,a) as I(s,a)= 1 2πi ( Z C 1 + Z C 2 + Z C 3 )z s− 1 g(z)dz. Then the entireness of I(s,a) on these three curves can be trivially verified by definition. For example, on C 2 , z ̸= 0 so we always have g(z)’s denominator 1− e z ̸= 0⇒ the integrand z s− 1 g(z) is entire⇒ 1 2πi R C 2 z s− 1 g(z)dz is entire. Likewise, this Theorem 2.2.5 enables the zeta function to be analytically extended to the whole complex planeC ′ except for s = 1; namely, we will get ζ (s,a) = Γ(1 − s)I(s,a)∀s ∈ C\{1}. So let’s form a corollary: Corollary2.2.6. ζ (s,a) = P ∞ n=0 1 (n+a) s can be extended to a new meromorphic function ζ (s,a) = Γ(1 − s)I(s,a) where I(s,a)= 1 2πi R C z s− 1 e az 1− e z dz, and (I)ζ (s,a) has a simple pole ats=1; (II)Res s=1 ζ (s,a)=1. 26 Proof. (I) Property 2.2.3 (III)⇒ζ (s,a) is analytic ats=2,3... Theorem 2.2.5 (II):I is entire Γ(1 − s) has poles at1,2,3...                  ⇒1 is the only pole ofζ (s,a). (II) Lets=:1. I(1,a)= 1 2πi Z C e az 1− e z dz (1) = 1 2πi Z C 2 e az 1− e z dz (2) =Res z=0 e az 1− e z = lim z→0 (z− 0) e az 1− e z = lim z→0 e az +aze az − e z =− 1, where in (1) the integrals alongC 1 andC 3 cancel, and (2) is by the residue theorem since0 is the isolated singularity of the function e az 1− e z . Hence lim s→1 (s− 1)ζ (s,a)=− lim s→1 (1− s)Γ(1 − s)I(s,a) =− lim s→1 Γ(2 − s)I(s,a)=Γ(1)=1 Alsoζ (1,a)=∞                  def ⇒Res s=1 ζ (s,a)=1. 2.3 Correspondingresultsforζ (s) Based on our foregoing results, now we give all the analogues of theorems above applied in the particular Riemann zeta function case. To prove them, all we need is to leta=:1 in the corresponding theorems for the Hurwitz zeta function. Property2.3.1. (I)ζ (s) converges absolutely inΩ= {s∈C:σ > 1}. 27 (II) ζ (s) converges uniformly in{s ∈ C : σ > 1+ϵ }∀ϵ > 0; moreover, it converges uniformly on every compact subset ofΩ . (III)ζ (s)isanalyticinΩ . Moreover,thes-derivativeζ ′ n (s)ofζ n (s):= P n k=1 1 k s convergesuniformlytoζ ′ (s) on every compact subset ofΩ . Theorem2.3.2. Γ( s)ζ (s)= Z ∞ 0 x s− 1 e − x 1− e − x dx≡ Z ∞ 0 x s− 1 e x − 1 dx∀σ > 1. Theorem2.3.3. (I)∀σ > 1,ζ (s) = Γ(1 − s)I(s), whereI(s) := 1 2πi R C z s− 1 g(z)dz andg(z) := e z 1− e z . (That is,I(s,a) = 1 2πi R C z s− 1 e z 1− e z dz.) HereC is still the infinite path shown in Figure 2.1. (II)I(s) is entire. Corollary2.3.4. ζ (s) = P ∞ n=1 1 n s can be extended to a new meromorphic function ζ (s) = Γ(1 − s)I(s) where I(s) = 1 2πi R C z s− 1 e z 1− e z dz, and (I)ζ (s) has a simple pole ats=1; (II)Res s=1 ζ (s)=1. 28 Chapter3 AnalyticContinuationII:FunctionalEquations In the previous chapter, we explored ways to analytically extend the zeta functions (Riemann zeta function and Hurwitz zeta function) by their connections with the gamma function. Now let’s investigate a new way to do that from another perspective: Riemann’s functional equation. 3.1 Dirichletetafunction In this section, let’s explore a series called theDirichletetafunctionη (s), which is an alternate series obtained by modifying the Riemann zeta function. So sometimes it is known as the alternating zeta function, also denoted asζ ∗ (s). Definition3.1.1 (Dirichlet eta function). ∀σ > 0, theDirichletetafunction is defined to be η (s):=1− 1 2 s + 1 3 s − 1 4 s +···≡ ∞ X n=1 (− 1) n− 1 1 n s . Property3.1.2. (I)η (s) is convergent inΩ ∗ :={s∈C:σ > 0}. (II)η (s) is analytic inΩ ∗ . 29 Proof. (I) This is a direct consequence of Dirichlet’s test in the complex plane. (II) We see η (s)= ∞ X n=1 (2n− 1) − s − (2n) − s . Setf n (s):=(2n− 1) − s − (2n) − s so f n (s)=(2n− t) − s | t=1 t=0 = Z 1 0 (2n− t) − s ′ t dt=s Z 1 0 (2n− t) − s− 1 dt ⇒|f n (s)|⩽|s| Z 1 0 (2n− t) − (x+1) dt (1) ⩽ |s| (2n− 1) x+1 wheres=:x+iy and (1) is because 1 (2n− t) x+1 >0 is an increasing function oft in[0,1] forℜs=x⩾0. Thus S 2N (s) (the partial sum)⩽ ∞ X n=1 |s| (2n− 1) x+1 =|s| ∞ X n=1 1 (2n− 1) x+1 which clearly converges. Likewise we can showS 2N+1 (s) also converges,∀N. Also |S 2N (s)− S N (s)| → 0, and we can verify S 2N (s) converges uniformly on sets of the form {s : |s| < R,x⩾ a > 0}. Hence, by Lemma 2.2.2, we get η (s) is analytic on the right-half plane since S 2N (s)→η (s) asN →∞. Now let’s explore the connection between the eta function and the zeta function: ∀σ > 1,ζ (s)− η (s)=2· 1 2 s +2· 1 4 s +2· 1 6 s +2· 1 8 s +··· =2· 1 2 s 1+ 1 2 s + 1 3 s + 1 4 s +··· =2 1− s ζ (s). Therefore, we conclude ζ (s)= 1− 2 1− s − 1 η (s) ∀σ > 1. (3.1) 30 Remark: Going back to the proof of Property 3.1.2 from another perspective, if we assume the results of Corollary 2.3.4 and this (3.1) hold, we can also immediately deduce the fact thatη (s) is analytic. Observing this equality, sinceζ (s) converges (absolutely) inΩ = {s∈C : σ > 1} by Property 2.3.1, we knowζ (s) on the left-hand side is defined only in Ω (before all the analytic continuation conducted in the previous chapter), while by Property 3.1.2 theη (s) on the right-hand side is defined in Ω ∗ ={s∈C: σ > 0} whereη (s) is analytic. Notice 1− 2 1− s − 1 on the right-hand side has a pole ats = 1, so (3.1) provides a way to analytically extend the domain ofζ (s) fromΩ toΩ ∗ −{ 1}. In other words: ζ (s)            = 1− 2 1− s − 1 η (s) ∀σ > 1 := 1− 2 1− s − 1 η (s) ∀0<σ ⩽1 (3.2) Here the first “ =” meansζ (s) is indeed equal to the right-hand side while the second “=” meansζ (s) is defined to be equal to the right-hand side so that it’s analytic. 3.2 Functionalequations Now thatζ (s) has been extended to the half-plane, our next goal naturally is to duplicate our process to “reflect” this result to the entire complex plane, which is what we’re going to do in this section. To deduce such an equation, let’s first introduce a famous formula. 3.2.1 Hurwitz’sformula First, we form two definitions and the corresponding properties necessary for deriving Hurwitz’s formula. 31 Definition3.2.1. (1) For givenr∈(0,π ), we define S(r):={C− B r (2nπi ):∀n∈Z}. (2) Recall that we defined I(s,a) = 1 2πi R C z s− 1 g(z)dz ≡ 1 2πi R C z s− 1 e az 1− e z dz in Theorem 2.2.5. Now we further define I N (s,a):= 1 2πi Z C(N) z s− 1 g(z)dz≡ 1 2πi Z C(N) z s− 1 e az 1− e z dz, whereN is an integer andC(N) represents the dog-bone-like contour shown in the followingz-plane: C(N) C 1 C 2 Figure 3.1: The “dog bone” contour This group of definitions has the following property: Property3.2.2. (I)g(z)= e az 1− e z (as defined in Theorem 2.2.5, with a∈(0,1]) is bounded inS(r). (II)∀σ < 0,theintegralalongC 2 (i.e.,theoutercircle)approachesto0asN →∞;hencelim N→∞ I N (s,a)= I(s,a). Proof. (I) Setz =:x+iy. It is obvious that |g(z)|= e az 1− e z = e ax |1− e z | ⩽ e ax |1− e x | . (3.3) (1) Motivated by the definition of S(r), we define T(r) to be as shown in the following figure: 32 Figure 3.2: T(r) That is,T(r):={z =x+iy :|x|⩽1,|y|⩽π, |z|⩾r}. T(r) is compact⇒g(z) is bounded inT(r) |g(z+2πi )|=|g(z)|        ⇒g is bounded in e T(r):={z :|x|⩽1,|z− 2nπi |⩾r,∀n∈Z}. (2) By (1), to proveg(z) is bounded in thisS(r), it suffices to show g(z) is bounded for|x|>1. (i)x>1: By Inequality (3.3) above,|g(z)|⩽ e ax |1− e x | where in this case the numerator0<e ax ⩽e x (sincea∈(0,1]) and the denominator|1− e x |=e x − 1, so |g(z)|⩽ e ax |1− e x | ⩽ e x e x − 1 = 1 1− 1 e x < 1 1− 1 e = e e− 1 33 (where the last inequality is due tox>1), meaningg(z) is bounded forx>1. (ii)x<− 1: Likewise,|g(z)|⩽ e ax |1− e x | where in this case the numerator0 < e ax < 1 and the denominator|1− e x | = 1− e x , so we have |g(z)|⩽ e ax |1− e x | < 1 1− e x < 1 1− 1 e = e e− 1 (where the last inequality is due tox<− 1), meaningg(z) is bounded forx<− 1. By (i) (ii),|g(z)|⩽ e e− 1 ∀|x|>1. By (1) (2),g(z) is always bounded inS(r). (II) As shown in the graph (Figure 3.1), onC 2 ,z =Re iθ ,θ ∈[− π,π ]. So z s− 1 = R s− 1 e iθ (s− 1) =R σ − 1 e − tθ ⩽R σ − 1 e π |t| . By (I), C 2 ⊆ S(r) (I) ⇒ “g(z) is bounded onC 2 ; that is,|g(z)|⩽C,∀z =Re iθ , for some constantC” ⇒ z s− 1 g(z) = z s− 1 |g(z)|⩽R σ − 1 e π |t| C ⇒| 1 2πi Z C 2 (N) z s− 1 e az 1− e z dz|⩽2πR · R σ − 1 e π |t| C =2πR σ e π |t| C→0 asR→∞ wheneverσ < 0. That is, the integral alongC 2 (i.e. the outer circle) approaches to0. Also, sinceI N →I+ 1 2πi R C 2 (N) z s− 1 e az 1− e z dz asN →∞, we concludeI N →I. Theorem3.2.3 (Hurwitz’s formula). ∀σ > 1,ζ (1− s,a)= Γ( s) (2π ) s e − πis 2 F(a,s)+e πis 2 F(− a,s) 34 whereF(a,s):= P ∞ n=1 e 2nπia n s . Proof. Puttings=:1− s in Property 3.2.2, we get lim N→∞ I N (1− s,a)=I(1− s,a)∀σ > 1. Here I N (1− s,a) def = 1 2πi Z C(N) z − s g(z)dz =− N X n=− N n̸=0 Res z=2nπi (z − s g(z)) (3.4) by Cauchy’s residue theorem. Note that2(N +1)πi is outside the region so it’s excluded from the sum, and by Figure 3.2 above, we haven̸=0. Computing the residue, we get Res z=2nπi (z − s g(z))=:R(n)= lim z→2nπi (z− 2nπi )z − s g(z) (1) = lim z→2nπi z− 2nπi 1− e z z − s e az =e 2nπia (2nπi ) − s lim z→2nπi z− 2nπi 1− e z (2) = − e 2nπia (2nπi ) s , where (1) is because2nπi are simple poles, and (2) is by the fact that lim z→2nπi z− 2nπi 1− e z = lim z→2nπi 1 − e z =− e − 2nπi =− 1. Substituting the residue into Equation (3.4), I N (1− s,a)=− " N X n=1 R(n)+ N X n=1 R(− n) # = N X n=1 e 2nπia (2nπi ) s + N X n=1 e − 2nπia (− 2nπi ) s (where(2nπi ) s =n s (2π ) s i s =n s (2π ) s e π 2 i s and likewise(− 2nπi ) s =n s (2π ) s e − π 2 i s ) 35 = 1 (2πe πi 2 ) s N X n=1 e 2nπia n s + 1 (2πe − πi 2 ) s N X n=1 e − 2nπia n s . SendingN →∞, I(1− s,a)= 1 (2πe πi 2 ) s ∞ X n=1 e 2nπia n s + 1 (2πe − πi 2 ) s ∞ X n=1 e − 2nπia n s (where we define P ∞ n=1 e 2nπia n s =:F(a,s) and so P ∞ n=1 e − 2nπia n s =F(− a,s)) = 1 (2πe πi 2 ) s F(a,s)+ 1 (2πe − πi 2 ) s F(− a,s)= 1 (2π ) s (e − πis 2 F(a,s)+e πis 2 F(− a,s)) By Theorem 2.2.5, ζ (1− s,a)=Γ( s)I(1− s,a)= Γ( s) (2π ) s (e − πis 2 F(a,s)+e πis 2 F(− a,s)). In this proof, we defined a function F(a,s) as the Dirichlet series P ∞ n=1 e 2nπia n s . We usually call this function theperiodiczetafunction. Obviously, this function is periodic (ina) with period1. In particular, F(1,s)=ζ (s). Moreover, by definition it is easy to verify this series absolutely converges for σ > 1. 3.2.2 Riemann’sfunctionalequation Theorem3.2.4 (Riemann’s functional equation). ∀s,ζ (1− s)=2(2π ) − s Γ( s)cos πs 2 ζ (s). 36 Proof. Puttinga=1 we get            F (1,s)=ζ (s), ζ (1− s,1)=ζ (s). So by Hurwitz’s formula (Theorem 3.2.3), ∀σ > 1,ζ (1− s)= Γ( s) (2π ) s e − πis 2 ζ (s)+e πis 2 ζ (s) = Γ( s) (2π ) s ζ (s) e − πis 2 +e πis 2 . Here e − πis 2 +e πis 2 =cos − πs 2 +isin − πs 2 +cos πs 2 +isin πs 2 =2cos πs 2 . So ζ (1− s)= Γ( s) (2π ) s ζ (s)2cos πs 2 . Equivalently, replacings with1− s in Theorem 3.2.4, we immediately get ζ (s)=2(2π ) s− 1 Γ(1 − s)sin πs 2 ζ (1− s). (3.5) This functional equation was first proposed by Euler in 1749, but it was finally named after Riemann since he extended this equation to the entire complex plane in his 1859 paper. In fact, in the previous section, we have analytically extendedζ (s) from{ℜs>1} to{ℜs>0:s̸=1} (namely, the entire right half-plane minus the singularity1), and now this functional equation enables us to further analytically extend it to the entire plane through a perspective different from the previous chapter. 37 According to preliminary observations, many strange features appear here. For example, ζ (− 1)≡ 1+ 1 2 − 1 + 1 3 − 1 +...≡ 1+2+3+... (3.5) = 2 − 1 π − 2 Γ(2)sin − π 2 ζ (2)≡− 1 12 , whereζ (2)= π 2 6 will be proved and further discussed later in the thesis; that is, 1+2+3+...=− 1 12 . Here even more famous than the Riemann zeta function is the fact that the infinite sum of all positive inte- gers actually equals anegative number. Under no circumstances is it likely to occur, but if we identify the previously introducedz(s) andζ (s) in §2.1.3, this phenomenon is afforded real mathematical significance. We also notice that takings = 1 2 gives uss = 1− s, which means the “ζ (s)” parts on both sides of (3.5) just disappear yet leaving only with an equation with respect to theΓ function. Therefore, although there is no further evidence, intuitively, this vertical lineℜs = 1 2 consisting of the points = 1 2 may play a critical role in the extended domain of the zeta function. From the functional equation, we can further deduce: Corollary3.2.5. Φ( s)=Φ(1 − s) whereΦ( s):=π − s 2 Γ s 2 ζ (s). To prove this corollary, let’s first introduce two famous formulas about the gamma function in the following lemma: 38 Lemma3.2.6. (I) (Legendre’s duplication formula) Γ(2 s)= 2 2s− 1 √ π Γ( s)Γ( s+ 1 2 ). (II) (Euler’s reflection formula) Γ( s)Γ(1 − s)= π sinπs . Proof. (I) By definition of the beta function, we have B(p,q)= Γ( p)Γ( q) Γ( p+q) = Z 1 0 x p− 1 (1− x) q− 1 dx. (3.6) (3.6) p=:q=:s = ==== ⇒ x=: 1+u 2 B(s,s)= Γ( s)Γ( s) Γ(2 s) = 1 2 2s− 1 Z 1 − 1 1− u 2 s− 1 du ⇔2 2s− 1 Γ( s) 2 =Γ(2 s)· 2 Z 1 0 1− u 2 s− 1 du. (3.7) Likewise, (3.6) x=:u 2 = === ⇒B(p,q)=2 Z 1 0 u 2p− 1 1− u 2 q− 1 du. So in the right-hand side of Equation (3.7), the factor 2 Z 1 0 1− u 2 s− 1 du=B 1 2 ,s = Γ 1 2 Γ( s) Γ 1 2 +s . Thus (3.7) becomes 2 2s− 1 Γ( s)Γ s+ 1 2 = √ π Γ(2 s); that is, 39 Γ(2 s)= 2 2s− 1 √ π Γ( s)Γ s+ 1 2 . (II) This formula can be proved by various methods, such as the Weierstrass definition of the gamma function or the contour integral; please refer to [1]. The reason (I) is called the Legendre duplication formula is that it gives a method for calculatingΓ(2 s) byΓ( s). And in fact, this formula can be generalized to Γ( s)Γ s+ 1 m ··· Γ s+ m− 1 m =(2π ) m− 1 2 m 1 2 − ms Γ( ms). (See Chapter 12 of [2].) Now let’s turn to the proof of Corollary 3.2.5. Proof. From Lemma 3.2.6, we have Lemma 3.2.6 (I) s=: 1− s 2 ⇒ 2 s π 1/2 Γ(1 − s)=Γ 1− s 2 Γ 1− s 2 Lemma 3.2.6 (II) s=: s 2 ⇒ Γ s 2 Γ 1− s 2 = π sin πs 2        cancelΓ(1 − 2/s) =========⇒ Γ s 2 sin πs 2 π = Γ 1− s 2 2 s π 1 2 Γ(1 − s) ⇔Γ(1 − s)sin πs 2 = Γ 1− s 2 π 1 2 2 s Γ s 2 . By the functional equation (3.5), ζ (s)=2(2π ) s− 1 Γ 1− s 2 π 1 2 2 s Γ s 2 ζ (1− s) 40 ⇔ζ (s)Γ s 2 =ζ (1− s)Γ 1− s 2 π s− 1 2 ⇔Φ( s)=Φ(1 − s). 3.2.3 Hurwitz’sfunctionalequation In the previous chapter, we extended the Riemann zeta function to the Hurwitz zeta function and deduced both of their expressions of analytic continuation. Now we will similarly extend Riemann’s functional equation to all Hurwitz’s zeta functions. Theorem3.2.7. For1⩽h⩽k, we have ζ 1− s, h k = 2Γ( s) (2πk ) s k X r=1 cos πs 2 − 2πrh k ζ s, r k . Proof. In the periodic zeta function (as defined in Theorem 3.2.3), setting a=: h k we obtain F h k ,s = ∞ X n=1 e 2πinh/k n s . Now write n=:qk+r(1⩽r⩽k andq∈N={0,1,2,...}). 41 So F h k ,s = k X r=1 ∞ X q=0 e 2πirh/k (qk+r) s = 1 k s k X r=1 e 2πirh/k ∞ X q=0 1 q+ r k s =k − s k X r=1 e 2πirh/k ζ s, r k . Likewise, F − h k ,s =k − s k X r=1 e − 2πirh/k ζ s, r k . Therefore, takinga=: h k in Hurwitz’s formula (Theorem 3.2.3), ζ 1− s, h k = Γ( s) (2π ) s e − d(s) F h k ,s +e d(s) F − h k ,s (whered(s):= πis 2 ) = Γ( s) (2π ) s e − d(s) k − s k X r=1 e 2πirh/k ζ s, r k +e d(s) k − s k X r=1 e − 2πirh/k ζ s, r k ! = Γ( s) (2πk ) s k X r=1 e − πis/ 2 e 2πirh/k +e πis/ 2 e − 2πirh/k ζ s, r k = 2Γ( s) (2πk ) s k X r=1 cos πs 2 − 2πrh k ζ s, r k . In particular, Theorem 3.2.7 h=:k=:1 = ====⇒ Theorem 3.2.4. 3.3 AnelementaryproofofRiemann’sfunctionalequation In this last section of the chapter, we provide an alternative proof of Theorem 3.2.4 which avoids all complex analysis tools but only resorts to elementary approaches. 42 3.3.1 Jacobithetafunction Definition 3.3.1 (Jacobi theta function). We define ϑ(x) := P n∈Z e − πn 2 x , which is called the Jacobi thetafunction. For this function we have Property3.3.2. ϑ(x)= r 1 x ϑ( 1 x ). (3.8) Proof. Poisson summation formula says X n∈Z f(n)= X k∈Z Z +∞ −∞ f(y)e − 2πiky dy. The proof of this formula uses Fourier transforms; details can be referred to [47] or [35]. Takingf(n)=:e − πn 2 x , we get X n∈Z e − πn 2 x = X k∈Z Z +∞ −∞ e − πy 2 x e − 2πiky dy = X k∈Z e − πk 2 x Z +∞ −∞ e − πx (y+ ik x ) 2 dy (1) = X k∈Z e − πk 2 x Z +∞ −∞ e − πxz 2 dz (2) = X k∈Z e − πk 2 x r 1 x where (1) is obtained by lettingz :=y+ ik x , and (2) is by the Gaussian integral: R +∞ −∞ e − x 2 dx= √ π . Hence by definition, it is equivalent to ϑ(x)= r 1 x ϑ( 1 x ). 43 This property was first obtained by Cauchy through Fourier analysis, while Jacobi also gave its proof using elliptic functions. Now we denoteψ (x):= P ∞ n=1 e − πn 2 x , which was originally given by Riemann in the 1859 paper [37]. Obviously, we have ϑ(x)=1+2 ∞ X n=1 e − πn 2 x =1+2ψ (x). Combining Equation (3.8), we immediately get 1+2ψ (x)= r 1 x 1+2ψ ( 1 x ) ; that is, ψ (x)= 1 √ x ψ ( 1 x )+ 1 2 √ x − 1 2 . (3.9) 3.3.2 RelationbetweenΓ( s)andζ (s) The Gamma function is defined as Γ( s)= R +∞ 0 x s− 1 e − x dx, so Γ s 2 def = Z +∞ 0 x s 2 − 1 e − x dx. In order to construct the form ofψ (x) in (3.9), we setx=:πn 2 t, so Γ s 2 = Z +∞ 0 π s 2 n s t s 2 − 1 e − πn 2 t dt Hence π − s 2 n − s Γ s 2 = Z +∞ 0 t s 2 − 1 e − πn 2 t dt. 44 Now taking summation fromn=1 to∞, π − s 2 Γ s 2 ∞ X n=1 1 n s = ∞ X n=1 Z +∞ 0 t s 2 − 1 e − πn 2 t dt= Z +∞ 0 t s 2 − 1 ∞ X n=1 e − πn 2 t dt, where the integral and sum are interchangeable due to the uniform convergence of the sum P ∞ n=1 e − πn 2 t . That is, by definition of ψ (t) andζ (s), Φ( s)≡ π − s 2 Γ s 2 ζ (s)= Z +∞ 0 t s 2 − 1 ψ (t)dt. (3.10) We split the right-hand-side integral as R 1 0 t s 2 − 1 ψ (t)dt+ R +∞ 1 t s 2 − 1 ψ (t)dt, where Z 1 0 t s 2 − 1 ψ (t)dt= Z 1 0 t s 2 − 1 1 √ t ψ 1 t + 1 2 √ t − 1 2 dt= Z 1 0 t s− 3 2 ψ 1 t dt+ 1 s(s− 1) (Setting 1 t =:u) = Z +∞ 1 u − s+1 2 ψ (u)du+ 1 s(s− 1) . Hence the right-hand side of Equation (3.10) becomes Z +∞ 0 t s 2 − 1 ψ (t)dt= Z 1 0 t s 2 − 1 ψ (t)dt+ Z +∞ 1 t s 2 − 1 ψ (t)dt = Z +∞ 1 u − s+1 2 ψ (u)du+ 1 s(s− 1) + Z +∞ 1 t s 2 − 1 ψ (t)dt = Z +∞ 1 t − s+1 2 +t s 2 − 1 ψ (t)dt+ 1 s(s− 1) . Thus (3.10) becomes Φ( s)≡ π − s 2 Γ s 2 ζ (s)= Z +∞ 1 t − s+1 2 +t s 2 − 1 ψ (t)dt+ 1 s(s− 1) . (3.11) 45 Settings=:1− s, Φ(1 − s)≡ π − 1− s 2 Γ 1− s 2 ζ (1− s)= Z +∞ 1 t s 2 − 1 +t − s+1 2 ψ (t)dt+ 1 s(s− 1) . (3.12) Comparing these last two equations we find the right-hand sides are equal, so we also conclude π − s 2 Γ s 2 ζ (s)=π − 1− s 2 Γ 1− s 2 ζ (1− s); (3.13) namely, Φ( s)=Φ(1 − s). 3.3.3 FromΦ( s)toRiemann’sfunctionalequation In Equation (3.13) we just proved, in order to apply the Legendre duplication formula let’s multiply both sides byΓ s+1 2 , which gives us π − s 2 Γ s 2 Γ s+1 2 ζ (s)=π − 1− s 2 Γ 1− s 2 Γ s+1 2 ζ (1− s). Notice here        Lemma 3.2.6 (I) s:=2s ⇒ Γ s 2 Γ s+1 2 = √ π 2 s− 1 Γ( s) Lemma 3.2.6 (II) s=: s+1 2 ⇒ Γ s+1 2 Γ 1− s 2 = π cos πs 2 Thus π − s 2 √ π 2 s− 1 Γ( s)ζ (s)=π 1+s 2 π cos πs 2 ζ (1− s) ; that is, ζ (1− s)=2 1− s π − s cos πs 2 Γ( s)ζ (s). 46 Setting1− s=:s, we get ζ (s)=2 s π s− 1 cos π (1− s) 2 Γ(1 − s)ζ (1− s). Equivalently, ζ (s)=2 s π s− 1 sin πs 2 Γ(1 − s)ζ (1− s), which is the same as Equation (3.5). Hence we now have completed another way to deduce Riemann’s functional equation (Theorem 3.2.4). 47 Chapter4 ValuesofZetaFunctionsatIntegers 4.1 Baselproblem Let’s start with the easiest valueζ (2)=1+ 1 2 2 + 1 3 2 + 1 4 2 +··· which is the summation of the reciprocals of the squares of all positive integers. By elementary analysis this is a convergent series, meaning it has a specific value less than infinity. What is it? This problem is called the Basel problem, which was first proposed by Pietro Mengoli in 1644. Here Basel is the third-largest Swiss city where Jakob Bernoulli taught mathematics; Bernoulli’s work raised this issue to the public’s attention, leading to the name “Basel problem”. ([10]) Even this seemingly simple problem had remained unsolved for more than 90 years until Euler [12] obtained the result in 1735, which is a surprising π 2 6 . Here π is the well-known mathematical constant 3.14159265..., the ratio of a circle’s circumference to its diameter; the reason why it is surprising is that π unexpectedly appears in a place without the presence of any circles, and this phenomenon was quite rare in Euler’s era. In fact, when giving this value, Euler further deduced: ζ (4)≡ 1+ 1 2 4 + 1 3 4 + 1 4 4 +...= π 4 90 , ζ (6)≡ 1+ 1 2 6 + 1 3 6 + 1 4 6 +...= π 6 945 , 48 ··· , until the value of ζ (26)≡ 1+ 1 2 26 + 1 3 26 + 1 4 26 +··· Indeed, Euler’s generalization of the Basel problem in his proof, which further clarified the nature of the Riemann zeta function, provided ideas and a basis for Riemann’s paper later in 1859. Next, let’s introduce an explicit method to solve the Basel problem: By the Taylor series expansion, for allx∈(−∞ ,+∞): sinx=x− x 3 3! + x 5 5! − x 7 7! +··· So sinx x =1− x 2 3! + x 4 5! − x 6 7! +··· (4.1) Since the solution set of the equation sinx x =0 is{x:x=nπ,n ∈Z−{ 0}}, Euler hypothesized sinx x = 1− x π 1+ x π 1− x 2π 1+ x 2π ··· = 1− x 2 π 2 1− x 2 4π 2 1− x 2 9π 2 ··· (4.2) By this equation, the coefficient of x 2 is 1 π 2 + 1 4π 2 + 1 9π 2 +··· =− 1 π 2 ∞ X n=1 1 n 2 . Comparing coefficients of x 2 in (4.1) and (4.2), we obtain ∞ X n=1 1 n 2 = π 2 6 . 49 It is noteworthy that Euler did not prove the validity of Equation (4.2) but directly assumed that it held. Not until a century later did Weierstrass prove this equation using the Weierstrass factorization theorem. Lastly, we present another method by resorting to a Fourier series as follows. Expandf(x)=x 2 ,x∈[− π,π ] to the Fourier series: f(x)= π 2 3 +4 ∞ X n=1 (− 1) n cosnx n 2 . Letx=π so we directly obtain ∞ X n=1 1 n 2 = π 2 6 . 4.2 FromBernoullinumberstoBernoullipolynomials As we will later explore, values of ζ (s) are related to Bernoulli numbers B n , which are defined as the coefficients in the following power series: z e z − 1 = ∞ X k=0 B k k! z k . (4.3) As the name suggests, Bernoulli numbers were originally introduced by Jacob Bernoulli in 1713. But notice that sometimes (for example, in topology or algebra) we have another different definition of Bernoulli numbers, which we denote as f B k in this thesis. They are defined as the following coefficients: z e z − 1 =1− z 2 + ∞ X k=1 (− 1) k+1 f B k z 2k (2k)! where|z|<2π. (4.3 ′ ) 50 Thus we have                      B 0 =1,B 1 =− 1 2 B 2k =(− 1) k+1 f B k ∀k⩾1 B 2k+1 =0∀k⩾1 (4.4) Also note that sometimes we use another convention ofB 1 defined as positive 1 2 , denotedB + 1 := 1 2 . Not only are they both called the same name, they actually share the same notation f B k without the differen- tiating tilde above in reality. In fact, numerous notation abuses like this can be found in math. The most notorious perhaps are inclusion symbols. Modern math identifies the pair (⊂ ,⊆ ) with(⊊ ,⊂ ). 1 That di- rectly causes that “⊂ ” has two possible meanings, which is confusing but remains so till today. To avoid such unnecessary ambiguity, here we always use (⊊ ,⊆ ) to denote “proper subset” and “subset” in this thesis, because this pair is always well-defined and unequivocal. What’s interesting is similar notations (<,⩽) don’t have any ambiguous definitions. Indeed, the first pair (⊂ ,⊆ ) was designed to be analogous to (<,⩽). Except for certain historical coincidences, the reason also includes that mathematicians care about “=” for numbers much more than for sets, since “set” is a much broader construction than “num- ber”. Another reason is that not any two sets can be endowed with an inclusion relation, while any two real numbers can have a size relation sinceR is a totally ordered set in general. 2 In this chapter, we adopt the first definition, Equation (4.3), in all the chapters before the K-theory chapter, which is frequently used in number theory. We will turn to the other definition, which is more broadly used in algebra and topology, when we discuss algebraic K-theory later in this thesis. The reason many topologists would like this unsigned definition of B k is that topology deals with negative numbers much less than number theory does, and using a signed B k would require us to consider splitting its negative sign each time when we only need its positive part. As a generalization of Bernoulli numbers, we define: 51 Definition4.2.1 (Bernoulli polynomials). ∀x∈C, the Bernoulli polynomialsB n (x) are defined by ze xz e z − 1 = ∞ X n=0 B n (x) n! z n , where|z|<2π. In particular,B n (0)=B n . We have the following properties aboutB n (x) andB n : Property4.2.2. (I)B n (x+1)− B n (x)=nx n− 1 ifn⩾1. Inparticular,puttingx=:0givesB n ≡ B n (0)=B n (1) ifn⩾2. (II)B n (x)= n X k=0 n k B k x n− k . (III)B n = n X k=0 n k B k . Proof. (I) ∞ X n=0 B n (x+1)− B n (x) n! z n = ∞ X n=0 B n (x+1) n! z n − ∞ X n=0 B n (x) n! z n def = z e (x+1)z e z − 1 − z e xz e z − 1 =ze xz Taylor === ∞ X n=0 x n n! z n+1 = ∞ X n=1 x n− 1 (n− 1)! z n . Equating coefficients of z n , B n (x+1)− B n (x)=nx n− 1 ∀n⩾1. (II) P ∞ n=0 Bn(x) n! z n def = z e z − 1 e xz z e z − 1 def = P ∞ n=0 Bn n! z n e xz Taylor === P ∞ n=0 x n n! z n                  ⇒ ∞ X n=0 B n (x) n! z n = ∞ X n=0 B n n! z n ! ∞ X n=0 x n n! z n ! 52 ⇒ B n (x) n! = n X k=0 B k k! x n− k (n− k)! by equating coefficients of z n ⇒B n (x)= n X k=0 B k x n− k n! k!(n− k)! = n X k=0 B k x n− k n k . (III) Puttingx=:1 in (II), B n (1)= n X k=0 n k B k (I) =B n (0)=B n . Property 4.2.2 provides a great way to compute the values ofB k : For example, B 0 def = 1 (III) ⇒ B 1 = 1 X k=0 1 k B k = 1 0 B 0 + 1 1 B 1 ⇒B 1 =− 1 2 . For the same reason, we obtain by deduction: B 2 = 1 6 B 3 =0 B 4 =− 1 30 B 5 =0 B 6 = 1 42 B 7 =0 B 8 =− 1 30 B 9 =0 B 10 = 5 66 B 11 =0 ··· ··· In the array above, we observe that values in the second column are all zero. So by induction, we easily obtain: 53 Corollary4.2.3. ∀n∈N + ,B 2n+1 =0. Likewise by (2), we deduce: B 0 (x)=1, B 1 (x)=x− 1 2 , B 2 (x)=x 2 − x+ 1 6 , B 3 (x)=x 3 − 3 2 x 2 + 1 2 x, B 4 (x)=x 4 − 2x 3 +x 2 − 1 30 , ... In conclusion, n B n B n (x) 0 1 1 1 − 1 2 x− 1 2 2 1 6 x 2 − x+ 1 6 3 0 x 3 − 3 2 x 2 + 1 2 x 4 − 1 30 x 4 − 2x 3 +x 2 − 1 30 5 0 x 5 − 5 2 x 4 + 5 3 x 3 − 1 6 x 6 1 42 x 6 − 3x 5 + 5 2 x 4 − 1 2 x 2 + 1 42 ... ... ... Table 4.1: Bernoulli numbers and polynomials 4.3 Valuesofzetafunctionsatnegativeintegers As a direct consequence of Corollary 2.2.6, we have Theorem4.3.1. ∀n∈N,ζ (− n,a)=− B n+1 (a) n+1 . 54 Proof. Corollary 2.2.6 s=:− n = === ⇒ζ (− n,a)=Γ(1+ n)I(− n,a)=n!I(− n,a), where I(− n,a)=Res z=0 z − n− 1 e az 1− e z . (4.5) Here we find z − n− 1 e az 1− e z =− z − n− 2 ze az e z − 1 def =− z − n− 2 ∞ X m=0 B m (a) m! z m . So (4.5) becomes I(− n,a)=Res z=0 − z − n− 2 ∞ X m=0 B m (a) m! z m ! =− B n+1 (a) (n+1)! by computing the residue. Consequently, ζ (− n,a)=− n! B n+1 (a) (n+1)! ≡− B n+1 (a) n+1 . In particular, by Property 4.2.2 (I), ζ (− n)=ζ (− n,1)=− B n+1 (1) n+1 =− B n+1 n+1 . (4.6) Hence by Corollary 4.2.3, we immediately get ∀n∈N + ,ζ (− 2n)=0. This means all negative even integers are zeros of the Riemann zeta function. Another way to deduce this equation, which is much more direct, is by taking s =: 2n+1 in Theorem 3.2.4 (Riemann’s functional equation). We will find the factor cos (2n+1)π 2 vanishes while all the other factors are finite, for any n∈N + . 55 These zeros are very easy to be clarified, so we call them the trivialzeros ofζ (s). Notice the zeros of ζ (s) here all come from zeros of the cosine function, so these trivial zeros are all simple zeros. Consequently, now we have clarified the values of the Hurwitz (thus, Riemann) zeta function at all negative integers. By the deduction formula provided in Property 4.2.2, we are able to compute all Bernoulli polynomials (numbers) whereby we may compute the exact values ofζ (− n,a) (and thusζ (− n)). 4.4 Valuesofzetafunctionsatnonnegativeintegers When we try putting n =: 0 in (4.6), we’ll get ζ (0) = − B 1 . But as introduced earlier, values of B 1 has two conventions: one is B + 1 = 1 2 and the other is B − 1 = − 1 2 . So in order to get a well-defined value of ζ (0), let’s try using the functional equation. First Corollary 2.3.4 saysζ (s) has a simple pole ats=1 with residue1, meaninglim s→1 (s− 1)ζ (s)=1. Motivated by this, we have (3.5) × (1− s) =⇒ (1− s)ζ (s)=2 s π s− 1 [(1− s)Γ(1 − s)]sin πs 2 ζ (1− s) lim s→1 =⇒            LHS≡ lim s→1 (1− s)ζ (s)=− 1 RHS≡ lim s→1 2 s π s− 1 [(1− s)Γ(1 − s)]sin πs 2 ζ (1− s)=2· 1· 1· 1· ζ (0)=2ζ (0) ⇔ζ (0)=− 1 2 . (4.7) So what about its values at positive integers? As we saw in §3.2.2, the Riemann functional equation offers a kind of symmetry for ζ (s). So let’s try takings=:2n in Theorem 3.2.4: ζ (1− 2n)=2(2π ) − 2n Γ(2 n)cos(πn )ζ (2n). 56 By (4.6),ζ (1− 2n)=− B 2n 2n , so − B 2n 2n =2(2π ) − 2n (2n− 1)!(− 1) n ζ (2n). That is, ζ (2n)=(− 1) n+1 (2π ) 2n B 2n 2(2n)! ∀n∈N + . (4.8) This equation provides all values of ζ (s) at positive even integers. By this equation, in turn, we can obtain some new properties of Bernoulli numbers: Corollary4.4.1. (1)(− 1) n+1 B 2n >0; i.e., the Bernoulli numbersB 2n alternate in sign. (2)(− 1) n+1 B 2n ∼ 2(2n)! (2π ) 2n asn→∞. (3)|B 2n |→∞ asn→∞. Proof. (1) By (4.8),∀n ∈N + ,ζ (2n) def = P ∞ k=1 1 k 2n > 0 ⇒ (− 1) n+1 B 2n > 0, meaning B 2n alternate in sign. (2) (4.8)⇔(− 1) n+1 B 2n = 2(2n)!ζ (2n) (2π ) 2n . Obviously, asn→∞, ζ (2n) def = ∞ X k=1 1 k 2n ≡ 1+ 1 2 2n + 1 3 2n +···→ 1. So lettingn→∞ in (4.8) gives the result. (3) is a direct consequence of (2), since| 2(2n)! (2π ) 2n |→∞ asn→∞. 57 Naturally, the next goal is to explore values at positive odd integers. Let’s imitate our previous proce- dure and takes=:2n+1 again in the functional equation in Theorem 3.2.4: ζ (− 2n)=2(2π ) − 2n− 1 Γ(2 n+1)cos π (2n+1) 2 ζ (2n+1). Unfortunately, we find cos π (2n+1) 2 = 0 on the right-hand side, meaning all the information about ζ (2n+1) vanishes as the vanishing of this factor. So for odd integers, this method fails. Nonetheless, recall that this exactly same procedure gave us another result, ζ (− 2n) = 0∀n ∈ N + , as stated in the previous section. That really tells us that every cloud has a silver lining; some seemingly awful things also have a few blessings in disguise. Indeed, mathematicians haven’t yet found any relevant universal formulas that can represent the Rie- mann zeta function at positive even integers. As a result, they had to compute these values one by one. For example, Roger Apéry, a French mathematician at the age of sixty, finally proved that the sum of the reciprocals of all integer cubic numbers is an irrational number in 1978. (See [49].) So the value ζ (3)=1.20205690315... is calledApéry’sconstant. Moreover, we can build a connection between this constant and the derivatives of the gamma function. Let’s first introduce: Definition4.4.2. We call the followingm-th derivative thepolygammafunction: ψ (m) (z):= d m dz m ψ (z)= d m+1 dz m+1 lnΓ( z). In particular, puttingm=:0 we have ψ (z):=ψ (0) (z)= Γ ′ (z) Γ( z) . 58 ψ (z) is called thedigammafunction. Property4.4.3. ζ (3)=− 1 2 ψ (2) (1) def = − 1 2 Γ ′′′ (1)+ 3 2 Γ ′ (1)Γ ′′ (1)− Γ ′ (1) 3 . The proof of this property can be found on Page 96 in [44]. It is noteworthy that values ofζ (2n) are also irrational numbers but the difference is that we can relate these values with our familiar π and unify them together in one formula (4.8) with respect to Bernoulli numbers. Therefore, we have enough reason to conjecture that we can generalize Property 4.4.3 above and build a connection between allζ (2n+1) and the gamma function. Unluckily, no results like this have been discovered yet. Lastly, we attach a table of some values ofζ (s) at other positive odd integers: s ζ (s) 5 1.0369277551433699263... 7 1.0083492773819228268... 9 1.0020083928260822144... 11 1.0004941886041194645... 13 1.0001227133475784891... 15 1.0000305882363070204... ··· ··· Table 4.2: ζ (s) at positive odd integers 59 Chapter5 Zerosofζ (s) Values of the independent variables such thatζ (s) vanishes (namely, zeros ofζ (s)) also play a central rule in researching values of the Riemann zeta function. However, modern researches in this area are notably binary, in the sense that all zeros ofζ (s) outside of the region{s∈C:01, i.e.,ζ (s)̸=0,ℜs>1. Proof. Assume σ = ℜs > 1. From Euler’s product formula|ζ (s)| = Q p |1− p − s | − 1 and|1− p − s | ⩽ |1+p − σ |, we have |ζ (s)|⩾ Y p 1+p − σ − 1 =e − P p ln(1+p − σ ) ⩾e − P p p − σ =e − P(σ ) . The last inequality sign holds since for anyx>0,ln(1+x)<x. SinceP (s) converges in allℜs>1, we have |ζ (s)|⩾e − P(σ ) >0. This impliesζ (s)̸=0,ℜs>1, which completes the proof. Further, by the symmetry shown in the functional equation (Theorem 3.2.4), we immediately get the regionℜs < 0 is also a zero-free region except only for those trivial zeros s = − 2k = − 2,− 4,− 6,... arising from the vanishing ofcos( πs 2 ). That is: Corollary5.1.2. ζ (s)̸=0∀s∈{s:σ < 0,s̸=− 2k,k∈N + }. Concluding what we have discovered in Theorem 5.1.1 and Corollary 5.1.2, we have Theorem5.1.3. All nontrivial zeros ofζ (s) fall in the region0⩽ℜs⩽1. 61 So in conclusion, by far, the region where zeros ofζ (s) can exist has been restricted to the following RegionR: Figure 5.1: Zero-free region Let’s further explore Equation (3.2) in §3.1. If we set 0 < s < 1, we find on the right-hand side the factor 1− 2 1− s − 1 < 0 while the Dirichlet eta function is a positive alternating sum. Hence we have deduced: Theorem5.1.4. ζ (s)<0 ifs∈(0,1)⊂ R. That means the line{s∈R:0 1 is a zero-free region of ζ (s). Now by Property 5.2.2, we obtain the functionξ (s) has no zeros in the regionℜs > 1. By Equation (5.1) (ξ (s) = ξ (1− s)) we seeξ (s) also has no zeros in the regionℜs<0. In conclusion, we have: Theorem5.2.3. All zeros ofξ (s), that is, all non-trivial zeros ofζ (s), fall in the region0⩽ℜs⩽1. This theorem, of course, is equivalent to Theorem 5.1.3 as we deduced in the previous subsection. In fact, it was Riemann who obtained this result in his 1859 paper, except he did this by analyzing the function 64 ξ (t) as defined above and obtained that ξ (t) vanishes only in− i 2 ⩽ ℑt⩽ i 2 , which is equivalent to our previous theorem. By Equation (5.3) (ξ (t)=ξ (− t)), we find the zeros of ξ (t) are distributed symmetrically with respect to the origin in the region− 1 2 ⩽ℑt⩽ 1 2 (that is,0⩽ℜs⩽ 1). In other words, ift 0 := x 0 +iy 0 is a zero ofξ (t), then− t 0 =− x 0 − iy 0 must also be a zero ofξ (t). Therefore, by Property 5.2.2, we immediately obtain: Theorem5.2.4. The nontrivial zeros of ζ (s) are distributed symmetrically with respect to the point s = 1 2 in the region 0⩽ℜs⩽ 1. In other words, ifs 0 := ( 1 2 +σ 0 )+it 0 is a nontrivial zero ofζ (s), then( 1 2 − σ 0 )− it 0 must also be a nontrivial zero ofζ (s). This symmetry shown in Theorem 5.2.4 is often referred to as thereversesymmetry. By definition of ξ (s) it is easy to verifyξ (¯s)=ξ (s); thus we have Theorem5.2.5. Thenontrivialzeropointsofζ (s)aresymmetricallydistributedwithrespecttotherealaxisℜs=0. Inother words, ifs 0 := σ 0 +it 0 is a nontrivial zero point ofζ (s), then ¯s 0 = σ 0 − it 0 must also be a nontrivial zero ofζ (s). The symmetry shown in Theorem 5.2.5 is often referred to as thereflectionsymmetry . Combining Theorem 5.2.4 and Theorem 5.2.5, we get Theorem5.2.6. The real part of the nontrivial zeros ofζ (s) distributed symmetrically with respect toℜs= 1 2 . The axis of symmetry hereℜs= 1 2 is called thecriticalline. This theorem above has only three cases: The non-trivial zeros ofζ (s) (if they exist) (i) all symmetrically distribute on both sides of the critical line; 65 (ii) distribute partially on both sides of the critical line and partially on the critical line; (iii) all distribute on the critical line. Having made this statement, Riemann presented the following result in a confident manner: Theorem5.2.7 ([37]). The number of roots ofξ (t) whose real part is between0 andT is approximately T 2π ln T 2π − T 2π . By our previous analysis, this result is equivalent to the fact that the number of zeros ofξ (s) (orζ (s)) in the region0⩽ℑs⩽T is approximately T 2π ln T 2π − T 2π . However, Riemann did not prove this theorem, but rather explained it in a few words. Evidently, he thought that the result was so simple that no proof was necessary, but he largely overestimated the mathematical ability of his readers. In fact, not until 1905 was this result proved by Mangoldt using contour integrals. Note that if we include the error term in the estimate, it can be written as T 2π ln T 2π − T 2π +O(lnT). This result is also known as theRiemann-Mangalt(R-M)formula. Using this formula, Riemann found the number of zeros ofξ (t) between0<ℜt<T is approximately equal to the number of its real roots in this region; in other words, the number of zeros of ζ (s) in 0 < ℑs < T is approximately equal to the number of zeros on the critical lineℜs = 1 2 . By this observation, Riemann conjectured the above-mentioned Case (iii) (below Theorem 5.2.6) holds: Conjecture5.2.8 (Riemann hypothesis). Allrootsofξ (t)=0arerealnumbers;thatis,allnontrivialzerosofζ (s)=0fallonthecriticallineℜs= 1 2 . 5.3 Valuesofζ (s)ontheliness =0ands =1 Can we further restrict the regionR? Yes, in this section we’ll exclude the two boundary liness = 0 and s = 1 from R. Historically, this result was considered much harder and therefore was obtained much later than the previous theorems. It was first proved by Hadamard [22] and de la Vallee Poussin [48] 66 independently in 1896 as a byproduct of attempting to prove the prime number theorem (PNT). First, by elementary algebra we have known this result: Lemma5.3.1. 3+4cosθ +cos(2θ )⩾0. Proof. 0⩽2(1+cosθ ) 2 =2cos 2 θ +4cosθ +2=3+4cosθ +cos(2θ ). Now let’s start with Equation (1.7) in §1.2: lnζ (s)= X p ∞ X n=1 1 np sn . By definition of complex logarithms, ln|ζ (s)|=ℜlnζ (s)= X p ∞ X n=1 cos(tlnp n ) np σn . Motivated by Lemma 5.3.1, we construct a polynomial this way: 3lnζ (σ )+4ln|ζ (σ +ti)|+ln|ζ (σ +2ti)| = X p ∞ X n=1 1 np σn (3+4cos(tlnp n )+cos(2tlnp n ))⩾0. Therefore, by taking the exponential we have∀σ > 1,t∈R, exp(3lnζ (σ )+4ln|ζ (σ +ti)|+ln|ζ (σ +2ti)|)≡ ζ (σ ) 3 |ζ (σ +it)| 4 |ζ (σ +2ti)|⩾1. 67 Concluding all results we have deduced, we have Lemma5.3.2. ∀σ > 1 andt∈R, ζ 3 (σ )|ζ (σ +ti)| 4 |ζ (σ +2ti)|⩾1. Then we may easily obtain the main theorem of this section: Theorem5.3.3. ζ (1+ti)̸=0∀t∈R. Proof. Sinceζ (1+0i)̸=0 is already known, it suffices to consider only t̸=0. By contradiction, suppose ∃t 0 ∈R−{ 0} such thatζ (1+t 0 i)=0. Then lettingσ →1 + , by the inequality in Lemma 5.3.2 we have lim σ →1 + ζ (σ ) 3 |ζ (σ +t 0 i)| 4 |ζ (σ +2t 0 i)|⩾1. (5.4) By Corollary 2.3.4ζ (s) has a simple pole ats=1, so the first factor ζ (σ ) 3 has apole of order3 at point 1. By assumption, the second factor|ζ (σ +t 0 i)| 4 has a zero of order at least4 at point1. That means the product of the first two factors ζ (σ ) 3 |ζ (σ +t 0 i)| 4 → 0 as σ → 1 + . Obviously, the last factor|ζ (σ +2t 0 i)|<∞∀t 0 ̸=0. Therefore, by the statements above we have lim σ →1 + ζ (σ ) 3 |ζ (σ +t 0 i)| 4 |ζ (σ +2t 0 i)|=0. (5.5) (5.4) and (5.5) contradict, which completes the proof. Again, the symmetry of Riemann’s functional equation gives us the following corresponding result: 68 Corollary5.3.4. ζ (ti)̸=0∀t∈R. Proof. For anyt∈R, sets=:ti in the functional equationΦ( s)=Φ(1 − s) whereΦ( s)=π − s 2 Γ s 2 ζ (s), so π − ti 2 Γ ti 2 ζ (ti)=π − 1− ti 2 Γ 1− ti 2 ζ (1− ti). By Theorem 5.3.3, the right-hand side̸= 0. Hence the left-hand side̸= 0, which impliesζ (ti) ̸= 0∀t ∈ R. After this section, we can now distill Figure 5.1 into: Figure 5.2: Refined zero-free region 69 5.4 Miscellaneousmodernresults 5.4.1 Bohr-Landautheorem Of course, if we can continue restrictingR to the lineℜs= 1 2 then we’ll be led to the proof of the Riemann hypothesis. What’s unfortunate is it’s extremely difficult for modern mathematicians to deeply research zeros in this mysterious area; that is why this mysterious area0<σ < 1 is called thecriticalstrip. More specifically, no one has actually found a way to show ζ (s)̸= 0 ins< 1− ϵ for any smallϵ . So far, all we can get is results like the following theorem: Theorem5.4.1. In the critical strip, there are no zeros of ζ (s) on the right side of the curveℜs = 1− C ln(|ℑ(s)|+2) for some C >0. This theorem was proved by Poussin [36] in 1899. Note that the curveℜs = 1− C/ln(|ℑ(s)|+2) in fact approachesℜs = 1 asℑs →∞, so this result is not of any great use to restrictR in the infinity perspective. In 1904, Bohr and Landau [6] showed another result about the distribution of zeros using Jensen’s formula. What they proved can be formed as the following theorem: Theorem5.4.2 (Bohr-Landau theorem). If the average value of|ζ (s)| 2 on linesℜs=σ is bounded forσ > 1 2 and uniformly bounded forσ ⩾ 1 2 +δ , then for anyδ > 0, the proportion of zeros that lie in the rangeℜs⩾ 1 2 +δ is infinitely small. In this theorem, “the average value of|ζ (s)| 2 on linesℜs = σ ” coincides with the usual definition of average values of a function over a domain, except we have to use a limit since we are considering an infinite domain. That is, lim T→∞ R T 1 |ζ (s)| 2 dt T − 1 (wheres=σ +it). 70 So in other words, what this theorem says is: ∀σ > 1 2 ,∃T 1 ,C 1 constants such that R T 1 |ζ (s)| 2 dt T− 1 <C 1 ∀T >T 1 ∃T 2 ,C 2 constants such that∀δ > 0 andσ ⩾ 1 2 +δ, R T 1 |ζ (s)| 2 dt T− 1 <C 2 ∀T >T 2        ⇒∀δ > 0, #nontrivial zeros in{ℜs⩾ 1 2 +δ, 0⩽t⩽T} #nontrivial zeros in{ℜs⩾ 1 2 ,0⩽t⩽T} →0. That is to say, any domain (no matter how small) that includes the critical line contains most of the zeros ofζ (s). This is a milestone in proving the Riemann hypothesis. Even if it doesn’t directly show that the critical lineℜs = 1 2 contains the entire zero set ofζ (s), it shows that this line indeed plays a unique role in the distribution of the zeros. As we can see, the result of this theorem depends on the validity of the fact that the average value of |ζ (s)| 2 on linesℜs = σ is bounded for σ > 1 2 and uniformly bounded for σ ⩾ 1 2 +δ . Fortunately, this condition has also been proven true by Landau in 1908 (while the main idea of his proof is from Hardy and Littlewood [23]). Let’s give the proof forσ > 1; forσ ⩽ 1 the proof is much more laborious and can be found in Chapter 9 of [11]. Proof. ∀σ > 1,ζ (s)=ζ (s), so|ζ (σ +it)|=|ζ (σ − it)|. Hence the average lim T→∞ R T 1 |ζ (s)| 2 dt T − 1 = lim T→∞ 1 2T Z T − T |ζ (σ +it)| 2 dt. (5.6) Also we have |ζ (σ +it)| 2 =ζ (σ +it)ζ (σ − it)= X m X n 1 n σ +it · 1 m σ +it . Since the last double series is dominated by P m P n 1 n σ · 1 m σ , it is uniformly convergent so it is integrable termwise. Hence the limit in (5.6) becomes 71 lim T→∞ 1 2T Z T − T |ζ (σ +it)| 2 dt= lim T→∞ X m X n 1 n σ · 1 m σ 1 2T Z T − T m n it dt. Form=n, the coefficient of 1 n σ · 1 m σ = 1 n 2σ is 1 2T R T − T 1dt=1. Form̸=n, the coefficient of 1 n σ · 1 m σ is 1 2T Z T − T m n it dt= 1 2T Z T − T e itln m n dt = 1 2T Z T − T cos tln m n +isin tln m n dt = 2 2T Z T 0 cos tln m n dt = 1 T sin(T ln m n ) ln m n , which approaches0 asT →∞ sincesin(T ln m n ) is bounded. Hence the last limit lim T→∞ X m X n 1 n σ · 1 m σ 1 2T Z T − T m n it dt= X m X n lim T→∞ 1 n σ · 1 m σ 1 2T Z T − T m n it dt = X n 1 n 2σ =ζ (2σ ). It is easy to verify this approaching is uniform if σ ⩾ σ 0 > 1, and moreover, it still holds for any σ ⩾ σ ′ 0 > 1 2 , that is, forℜs⩾ 1 2 +δ ∀δ > 0. Thus we conclude that∀σ > 1, the averagelim T→∞ R T 1 |ζ (s)| 2 dt T− 1 of|ζ (s)| 2 onℜs = σ is justζ (2σ ) and holds uniformly forℜs⩾ 1 2 +δ ∀δ > 0. That is the condition of Theorem 5.4.2. 5.4.2 Hardy’stheorem Although the Bohr-Landau Theorem in the last subsection shows the critical line is a “collection center” of all zeros, it actually cannot prove even one zero lies exactly on this line. In 1903 (the year before the Bohr-Landau theorem was proposed) we only knew the exact value of 15 zeros which, of course, are all on 72 the critical line. They were approximately computed by J. P. Gram [20] using the Euler-Maclaurin formula with the first ten zeros rounded to the fifth decimal and the last five zeros rounded to only the first decimal: Then-th zero Value 1 1/2+14.134725i 2 1/2+21.022040i 3 1/2+25.010856i 4 1/2+30.424878i 5 1/2+32.935057i 6 1/2+37.586176i 7 1/2+40.918720i 8 1/2+43.327073i 9 1/2+48.005150i 10 1/2+49.773832i 11 1/2+52.8i 12 1/2+56.4i 13 1/2+59.4i 14 1/2+61.0i 15 1/2+65.0i Table 5.1: Gram’s first 15 zeros In 1914, Backlund [4] proceeded the computation of zeros to the first 79 ones. Such computations are only limited to verifying finitely many zeros, but in the next year 1915, Hardy successfully proved a theorem that is involved in an infinite number of zeros on the critical line: Theorem5.4.3 (Hardy’s theorem). There are an infinite number of zeros that are located on the critical line ℜs = 1 2 . In other words, we have lim T→∞ N 0 (T)=∞ whereN 0 (T) denotes the number of zeros on the critical lineσ = 1 2 ,0<t⩽T. Proof. Let’s first recall, to obtain Equation (5.3) in §5.2, we did a transformation s=: 1 2 +ti from (5.1). To avoid possible confusion, here we rewrite ξ (t) as capitalized Ξ( t). For this even function, we can easily prove the following formula: Z ∞ 0 Ξ( t) t 2 + 1 4 cos(xt)dt= π 2 e x 2 − 2e − x 2 ψ e − 2x , 73 whereψ (s) := P ∞ n=1 e − n 2 πs is another form of the Jacobi theta function as introduced in §3.3.1. Details can be found in Chapter X of [46]. Puttingx =:− iα (α < π 4 ) and differentiating the formula above with respect to α for2n times, we get 2 π Z ∞ 0 Ξ( t) t 2 + 1 4 t 2n coshαtdt = (− 1) n cos 1 2 α 2 2n− 1 − 2 d dα 2n e 1 2 iα 1 2 +ψ e 2iα . In this equation, it is easily shown the last term on the left-hand side tends to 0 asα → π 4 (also see [46]), meaning we now have lim α → π 4 Z ∞ 0 Ξ( t) t 2 + 1 4 t 2n coshαtdt = (− 1) n π cos π 8 2 2n . (5.7) In §5.2, we have seen the zeros ofζ (s) on the lineℜs= 1 2 correspond exactly to the real roots ofΞ( t)=0. Hence, in the theorem “an infinite number of zeros being located on the critical line” is equivalent to “ Ξ( t) changing sign infinitely many times”. By contradiction, assume∃T > 0 such that Ξ( t) doesn’t change sign anymore for t > T . Thus the left-hand side of (5.7) is uniformly convergent (with respect toα ), meaning the limit and integral symbol will be interchangeable; that is, Z ∞ 0 Ξ( t) t 2 + 1 4 t 2n cosh( π 4 t)dt= (− 1) n π cos π 8 2 2n . Again, by the assumption, we have the left-hand side of the equation above does not change sign for increasingly large n, while the right-hand side obviously has alternating signs due to (− 1) n . This is a contradiction, which completes the proof. Three different proofs can be found in Chapter IX of [46]. The important position of this conclusion in researching zeros is self-evident, but it’s still very far from verifying RH since infinity is not an exact 74 number in math, and so we are still ignorant about zeros that don’t lie on the critical line in a quantitative way. We will explore these approaching results in the following subsection. 5.4.3 Approachingthecriticalline In 1921, Hardy and Littlewood [23] proved a theorem stronger than Hardy’s theorem. Theorem5.4.4 (Hardy-Littlewood theorem). ∃ constantK >0 andT 0 >0, such that∀T >T 0 ,N(T)⩾KT, whereN(T) is the number of zeros in the rectangle0⩽σ ⩽1, 0<t⩽T. The next remarkable result appeared in 1942 by Selberg [42], who further improved an integrand 2ξ (z)x z− 1 /z(z− 1) used in the proof of the Hardy-Littlewood theorem and obtained the following so- called critical line theorem: Theorem5.4.5 (Critical line theorem). ∃ constantT 0 >0, such that∀T >T 0 ,N(T)⩾KT lnT for someK >0. A breakthrough of this critical line theorem is that for the first time, this result showed that the ratio of zeros lying on the critical line to all zeros in the critical strip is a positive number (while this ratio in the Hardy-Littlewood Theorem is still zero); that is to say: Theorem5.4.6 (Critical line theorem ′ ). ∃ constantT 0 >0, such that∀T >T 0 ,N 0 (T)⩾A 2 N(T) for someA 2 >0. Followed by Selberg, Levinson [29] in 1974 further obtained the lower bound A 2 should be greater than one third; that is: Theorem5.4.7 (Levinson’s critical line theorem). ∃ constantT 0 >0, such that∀T >T 0 ,N 0 (T)⩾ 1 3 N(T). 75 In Levinson’s proof, he used a new method that combines the contribution of zeros ofζ (s) with that of zeros ofζ ′ (s). This method provides a new platform for subsequent mathematicians to further refine this constant. For example, nine years later, in 1983, Conrey [7] provedN 0 (T)⩾ .3685N(T); finally in 1989, he improved this constantA 2 to.4 in [8]: Theorem5.4.8 (Conrey’s critical line theorem). ∃ constantT 0 >0, such that∀T >T 0 ,N 0 (T)⩾ 2 5 N(T). 76 Chapter6 AlgebraicK-theoryandZetaFunctions In this chapter, we will present the connection between our zeta functions and algebraic K-theory by examining general K-groups of the ringZ of integers. Algebraic K-theory is a relatively new branch of algebra, which was first invented by Alexander Grothendieck in the study of algebraic varieties in the late 1950s. Indeed, in 1957, Grothendieck [21] only defined K 0 (X) of a variety X, and then subsequent generations of mathematicians did all the extension fromK 0 toK n in general. The details can be found in Appendix A and the first section below. So here we start with a brief introduction to general higher K-groupsK n . 6.1 ConstructionofhigheralgebraicK-theorygroupsK n One may find it confusing about the notations K 0 ,K 1 ,K 2 we defined in basic algebra (detailed definitions and properties can be found in Appendix A)− why should we put these numbers0,1,2 there? Intuitively, it is for the purpose of generalizing. Now that we’ve seen K 1 (R) = GL(R) ab ∼ = H 1 (GL(R),Z) and K 2 (R) ∼ = H 2 ([GL(R),GL(R)],Z), it is natural to ask if we can extend our K-constructions to higher K-groups? As we know, the derivative of a smooth function f(x) denotes the instantaneous rate of change of f(x) at any point x in its domain; after that, mathematicians wanted to explore further by duplicating 77 the same pattern to optimize their efforts, namely to examine “the rate of change of the rate of change of f(x)”, i.e., “the rate of change off ′ (x)”. That is how the second-order derivativesf (2) (x) came into being. What made it possible to extend the first-order derivatives is that f ′ (x) andf(x) are in the same category − that is, f ′ (x) is also a smooth function as is f(x). Lastly, by induction, all high-order derivatives are defined: f (n) (x):=[f (n− 1) (x)] ′ . Unfortunately, the same pattern cannot be directly used to defining general K n yet. IfK 1 were (iso- morphic to) H 1 (GL(R),Z) (which is true) while K 2 were H 2 (GL(R),Z) and if we could also prove K 0 ∼ = H 0 (GL(R),Z), then a natural extension: K n := H n (GL(R),Z) would directly be constructed. But the real definitions of K 0 andK 2 are not good (regular) enough, althoughK 1 andK 2 are somewhat connected byGL(R). Then again, that inspires us to find further unified relations equivalent to all our previous definitions, which connectK 0 ,K 1 andK 2 in a good pattern. We know that a famous result called theHurewiczthe- orem has connected together the two basic algebraic topology elements: homotopy theory and homology theory, via a map called theHurewiczhomomorphismh ∗ : π n (X)→ H n (X) for any path-connected space X and positive integer n. Now that the “homology” model doesn’t work well, motivated by this theorem, let’s try the “homotopy” model. To do that, we first introduce the higher homotopy groups in the next subsection. Of course, this is only one possible way to extend our previous K-groups toK n which has formed the basis of the modern K-theory, and there may be many other such good patterns to be explored. 6.1.1 Anintuitiveconstructionofthen-thhomotopygroup Now we will apply the topological extension model again to deduce an intuitive way of extendingπ 1 to π n . This definition of higher homotopy groups is crucial to understanding higher K-groups. Following the same pattern as the derivative example above, there is a group naturally related to every topological spaceX:π 1 (X;x 0 ):=(Ω x 0 X/≃,× ), whereΩ x 0 X :={α :[0,1]→X path;α (0)=α (1)=x 0 }, 78 ≃ identifies all path-homotopic paths, and × is defined as [α ]× [β ] := [α ∗ β ]. This is thefundamental group ofX (based atx 0 ). Again, the root of topological spaces is still our concrete metric spaces, so let’s first assume (X,d 0 ) is a metric space (with metricd 0 ) to get an intuitive idea. Like the process of extending first-order derivatives, if we can build a metric structure on Ω x 0 X thenΩ x 0 X andX will be in the same category (both are metric spaces), and then a natural extensionπ 1 (Ω x 0 X;c x 0 ) def = (Ω cx 0 (Ω x 0 X)/≃,× ) (where c x 0 ∈ Ω x 0 X is the constant path in X and Ω cx 0 (Ω x 0 X) def = “space of paths h : [0,1] → Ω x 0 X withh(0) = h(1) = c x 0 ”) will be made possible. Perhaps the only reasonable way to define a metric on Ω x 0 X is byd 1 (α,β ):=sup t∈[0,1] d 0 (α (t),β (t)) since the usually more natural “inf” is always0 here, and obviouslyd 1 satisfies the definition of metrics. In the metric space ( Ω x 0 X,d 1 ), we have: Property6.1.1 (Property of path homotopies). The pathsα andβ inX are path homotopic iff they belong to the same path-connected component of Ω x 0 X. In general, path homotopies are defined as follows, so we have to apply and verify this definition in the proof of this property. Definition6.1.2 (Path homotopy). Two pathsα 0 andα 1 are said to be path homotopic (denotedα 0 ≃α 1 ) if there exists a path homotopy betweenα 0 andα 1 , which means a continuous mapH :[0,1]× [0,1]→X such that                        H(s,0)=α 0 (s)∀s∈[0,1] H(s,1)=α 1 (s)∀s∈[0,1] H(0,t)=α 0 (0)=α 1 (0)∀t∈[0,1] H(1,t)=α 0 (1)=α 1 (1)∀t∈[0,1] . Before we prove this property, also note that by definition of path connectedness, α andβ belonging to the same path-connected component just means we can find a continuous function (called a path) h : [0,1]→Ω x 0 X such thath(0)=α andh(1)=β , i.e., the pathα can be smoothly pulled toβ inX (passing 79 through infinitely many paths which were characterized by h in the process) with the endpoints fixed and without passing through any “holes”. Proof. (⇒) First define H : [0,1]× [0,1] → X to be a path homotopy from α ∈ Ω x 0 X to β ∈ Ω x 0 X. ∀t∈[0,1], leth t ∈Ω x 0 X be the pathh t :[0,1]→X defined by h t (s):=H(s,t). Then under the metric d 1 we can directly verify h : [0,1] → Ω x 0 X defined by h(t) := h t is continuous so is a path in Ω x 0 X going from h(0) = α to h(1) = β , meaning α and β belong to the same path connected component of Ω x 0 X. (⇐) Now first define h : [0,1]→ Ω x 0 X to be a (continuous) path (inΩ x 0 X) going fromh(0) = α to h(1) = β ∈ Ω x 0 X. Let H : [0,1]× [0,1] → X be defined by H(s,t) := h t (s) where h t := h(t)∀t ∈ [0,1]. Then likewise we can directly verifyH is a continuous map and satisfies the definition of the path homotopy fromα toβ , meaningα ≃β . The proof above is trivial by directly applying a few definitions, but more significant is the fact that this so-called property can be considered what motivated us to define “path homotopic” in the first place before we topologically extended it to our usual Definition 6.1.2. That means, Property 6.1.1 is the real, intuitivedefinition of path homotopy as it should be, and Definition 6.1.2 is just a property which is metric- independent, so it was used as the “core property” as in the topological extension model. 1 Property 6.1.1 is equivalent to “the set consisting of path connected components ofΩ x 0 X”=π 1 (X;x 0 ). Hence likewise, “the set consisting of path connected components of Ω cx 0 (Ω x 0 X)”= π 1 (Ω x 0 X;c x 0 ), which inspires to formally define π 2 (X;x 0 ) := π 1 (Ω x 0 X;c x 0 ). As in the case of derivatives as well as most cases in math, once we achieve the hardest step from “1” to “2”, then the higher cases will become extremely immediate by induction: We define π n (X;x 0 ):=π n− 1 (Ω x 0 X;c x 0 ) forn⩾2, wherec x 0 ∈Ω x 0 X is the constant path (inX). That is,π n (X;x 0 ) =π n− 1 (Ω x 0 X;c x 0 ) =··· =π 1 (Ω n− 1 x 0 X;c n− 1 x 0 ) wherec n− 1 x 0 ∈ Ω n− 1 x 0 X is the constant 80 path (in Ω n− 2 x 0 X). As we did forn = 2 in the (⇒) part of the proof above, define H : [0,1] n → X, and likewise, we haveH(s 1 ,s 2 ,··· ,s n− 1 ,s n ) = (h(s n ))(s 1 ,s 2 ,··· ,s n− 1 ) = h sn (s 1 ,s 2 ,··· ,s n− 1 ) where h(s n )=h sn and(s 1 ,...,s n )∈[0,1] n (n=1,2,3,...). Now let’s further explore the other (⇐) part of the proof. Since every such functionh:[0,1]→Ω x 0 X is a path inΩ x 0 X, eachh(s) is the orbit (image) of a path inΩ x 0 X and thus all suchh’s form a set consisting of all paths in Ω x 0 X. Therefore π 2 (X;x 0 ) is formed by the homotopy classes of maps h. Also, we see there are infinitely many H’s (corresponding to theh’s as defined in this proof) over space X related to the definition of π 2 , but they have one property in common: allH’s sending the boundary of[0,1] 2 to the base pointx 0 , i.e., H(∂[0,1] 2 ) = x 0 . That is, as long as such the mapH sends the boundary of[0,1] 2 to the base pointx 0 , then such homotopy classes ofH satisfy our definition of π 2 (X;x 0 ) def = π 1 (Ω x 0 X;c x 0 ). So let’s summarize our exploration andinductively extend it ton-dimensions (which can be done immediately since then-th homology groups are also defined inductively as above): We haveπ n (X;x 0 ) =“the set of homotopy classes of mapsH : ([0,1] n ,∂[0,1] n )→ (X,x 0 )” i.e. = “the set of homotopy classes of mapsH :[0,1] n →X withH(∂[0,1] n )=x 0 ”. As we can see, the reason we care about this property above is that this property is metric-independent while it coincides with our original definition in metric spaces, which has enabled us to do another topo- logical extension to get rid of the restriction ofX being a metric space: Definition 6.1.3 (The n-th homotopy group). In a topological space X, we define the n-th homotopy group π n (X;x 0 ) :={homotopy classes of maps H : (I n ,∂I n ) → (X,x 0 ), where I := [0,1],x 0 ∈ X}. If we define [0,1] 0 as a point (because “0-dimentional” means a single point) and ∂[0,1] 0 :=∅, then in particular,π 0 (X;x 0 )= {path-components ofX}. Until now, we have completely demonstrated the entire process of defining the n-th homotopy groups π n (X;x 0 ) for general topological spacesX. We started with our familiar metric spaces looking for mo- tivations and defined π 2 (X;x 0 ). Then, motivated by the derivative example, we inductively constructed 81 π n (X;x 0 ) for metric spaces, followed by a topological extension which extended our constructions to all general topological spaces. Further, we can easily verify the following two properties (See Chapter 4 in [24]): Property6.1.4 (Properties ofn-th homotopy groups). (1)π n (X;x 0 ) is always abelian whenn>2. (2) We haveπ n ( Q α X α ) ∼ = Q α π n (X α ) ∀n, where Q α X α is a product of an arbitrary collection of path- connected spacesX α . 6.1.2 Classifyingspacesandplusconstruction Given a functionf, we want to know if there’s a function whose derivative isf? That is how its inverse operation,integration, came to be. Likewise, given any groupG, it is natural to ask whether we have a space whose fundamental group is G? Indeed, we can naturally construct such a connected topological spaceBG such thatπ 1 (BG)=G whileπ n (BG)=0∀n⩾2. Further constructions can be found in [30], and we callBG aclassifyingspace forG. In 1969, Michel Kervaire introduced the+-construction, which was adopted by Daniel Quillen in the algebraic K-theory: Definition6.1.5 ([40]). LetX be a connectedCW -complex with basepointx 0 chosen from the0-skeleton; letπ ⊴π 1 :=π 1 (X,x 0 ) be a perfect normal subgroup, meaningπ =[π,π ]=[π 1 ,π ]. Then we define X + as the new CW -complex obtained by attaching only 2-cells and 3-cells to X, so that the pair (X + ,X) satisfies: (1) Themapπ 1 (X,x 0 )→π 1 (X + ,x 0 ) induced by the inclusion is just the quotient mapπ 1 →π 1 /π. (2)∀π 1 /π moduleM (viewed as a local coefficient system on X andX + ), we haveH • (X + ,X;M)=0. Note that if the pair(X + ,X) satisfies condition (2), we call this pair homologicallyacyclic. 82 In Definition 6.1.5, the existence of X + defined this way is justified by more specific examinations of this construction. 2 And we can further proveX + is unique in the universal sense: that is, given any other CW complex ˜ X + containing X as a subcomplex while satisfying these same conditions, there exists a homotopy equivalenceX + → ˜ X + homotopic to the identity onX. We have the following property of the+-construction that we just constructed: Property6.1.6 ([50]). LetP beaperfectnormalsubgroupofgroupGwithcorresponding+-constructionf :BG→BG + .IfF(f) isthehomotopyfiberof f,thenπ 1 (F(f))istheuniversalcentralextensionofP andπ 2 (BG + ) ∼ =H 2 (P;Z). A proof can be found in [50], in which we applied Theorem A.3.3 in this thesis. Based on these definitions and theorems, we find, by uniqueness: π 1 BGL(R) + def = π 1 (BGL(R))/π def = GL(R)/π π =[π 1 (BGL(R)),π 1 (BGL(R))] ================ GL(R)/GL ′ (R); GL(R)/GL ′ (R) i.e. = GL(R)/E(R) def = K 1 (R). So we now have π 1 (BGL(R) + ) ∼ = K 1 (R). Before we draw any further conclusions, let’s examine π 2 (BGL(R) + ) andπ 0 (BGL(R) + ). In Property 6.1.6, letG=:GL(R) andP =:[GL(R),GL(R)]=E(R) as in theπ 1 case above, so we’ll immediately see: π 2 (BGL(R) + ) ∼ =H 2 (E(R);Z) Also we knowK 2 (R) ∼ =H 2 (E(R);Z)        ⇒π 2 (BGL(R) + ) ∼ =K 2 (R). However, in fact we have: π 0 (BGL(R) + ) def = “the set of path-components ofBGL(R) + ” BGL(R) + is path-connected ============== 1 (the trivial group), which is not equal toK 0 as expected. But it doesn’t mean this construction has failed; we don’t want to 83 give up all of our previous work, so let’s find a way to fix it. We want to further construct a space out of BGL(R) + whoseπ 0 isK 0 and hopefully whoseπ 1 ,π 2 are the same asπ 1 (BGL(R) + ) andπ 2 (BGL(R) + ), resp. Naturally we find π 0 (K 0 (R))× 1=π 0 (K 0 (R))× π 0 (BGL(R) + ) Prop6.1.4 ==== π 0 (K 0 (R)× BGL(R) + ), while π n (K 0 (R)× BGL(R) + ) = π n (BGL(R) + ) since K 0 (R) is naturally endowed with a discrete topology. That showsK 0 (R)× BGL(R) + is a “good” extension which both is “regular” and does agree with all of our previous definitions, hence providing us with a great way to finally define: Definition 6.1.7 (The n-th K-theory group). K n (R) := π n (K(R))∀n, where K(R) := K 0 (R) × BGL(R) + . Hence it is immediate that K n (R) = π n (BGL(R) + )∀n ⩾ 1. Furthermore, we have the following properties: Property6.1.8 ([40]). (1)K 3 (R) ∼ =H 3 (St(R),Z); (2)K i (R) ∼ =π i (BE(R) + ) ∀i⩾2; (3)K i (R) ∼ =π i (BSt(R) + ) ∀i⩾3. In this Property above, (1) is significant, because we already have K 1 (R) ∼ =H 1 (GL(R),Z),K 2 (R) ∼ = H 2 ([GL(R),GL(R)],Z), and now K 3 (R) ∼ = H 3 (St(R),Z), in which a sequence consisting of closely related spaces has been formed. 6.2 Restrictingtoringofintegers In this section, we shall only focus on the K-theory groups of the ring of integers, as they have a strong connection with zeta functions that will be introduced later. First, by basic algebraic K-theory as introduced in Appendix A, we shall deduce:K 0 (Z) ∼ =Z,K 1 (Z) ∼ = Z × = {− 1,1} ∼ = Z/2, and K 2 (Z) ∼ = Z/2 = {0,1}. Indeed, K 0 (Z) ∼ = Z follows directly from the 84 definition, either in the perspective of group completions by viewing Z as an abelian monoid or ofK 0 (Z)= K 0 (Iso(Z-Proj)) by viewingZ as a ring. As forK 1 (Z), let’s start with an example: Example6.2.1 (K 1 of a field) . For a (commutative) field F , we know every matrixA overF withdet(A)= 1 is a product ofe ij (r)’s by linear algebra; thusE n (F)⊇ SL n (F) soE n (F) = SL n (F)∀n⩾ 1 in this case. HenceE(F)=SL(F) def = kerdet. That means: SK 1 (F) def = SL(F)/E(F) is trivial by(A.1) =⇒ 1 → K 1 (F) → F × → 1 is exact⇔ K 1 (F) ∼ = F × i.e. ⇔ det/E:K 1 (F)→F × is an isomorphism. Therefore, we have obtained (by Whitehead’s lemma and the first isomorphism theorem) K 1 (F) = GL(F)/E(F)=GL(F)/SL(F) ∼ =F × . Moreover, we want to explore the first K-group of structures broader than a field (as described in Appendix §A.1.2 following the translated extension andPatternR): for example, for any Euclidean domain R (which isf 3 (B) constructed in Example 2.1.6), do we also haveK 1 (R) ∼ =R × ? Yes, we do. Theorem6.2.2 ([40],K 1 of a Euclidean ring). IfR is any Euclidean ring, thenSL n (R) = E n (R)∀n,SK 1 (R) vanishes (namely,SK 1 (R) is trivial), and K 1 (R) ∼ =R × . Shall we further extend this theorem and find more rings R such thatSK 1 (R) = 0? That motivated the following exploration by three K-theorists, providing an alternative way to find K 1 (Z): Theorem6.2.3 ([16], Bass-Milnor-Serre). SK 1 (O K ) = 0, whereO K is the ring of integers (as will further discuss in §6.3.2) in an algebraic number field K that we will explore further below. In particular, by definition O Q = Z (that’s why the elements ofZ are usually called the rational integers in algebraic number theory)⇒ SK 1 (Z) = 0 def ofSK 1 ⇐⇒ ker(det/E) = 0 ⇔ det/E is injective by(A.1) ⇐⇒ K 1 (Z) ∼ =Z × ={± 1}, which, of course, coincides with the previous Theorem 6.2.2. 85 Lastly, we computeK 2 (Z) by the following example: Example6.2.4. For any ringR, letx=(x 12 (1)x 21 (− 1)x 12 (1)) 4 ∈St(R). We have φ(x)=         1 1 0 1         1 0 − 1 1         1 1 0 1         4 =     0 1 − 1 0     4 =     1 0 0 1     =I ∈E(R) meaningx∈kerφ def = K 2 (R) ThA.3.3 ==== Z(St(R)). Indeed, as discussed in (A.4) in Appendix A, we have the following universal central extension ofE(R) that impliesK 2 (R) is precisely the center ofSt(R): 1→K 2 (R),→St(R) φ ↠ E(R)⊆ GL(R)↠ K 1 (R)→1. In particular, when R :=Z, we have x ∈ K 2 (Z). And it is not difficult to show that x has order 2 and generatesK 2 (Z) (see §4.2.19 in [40]), meaningK 2 (Z) is cyclic of order2 soK 2 (Z)=Z/2. Next, let’s start exploring higher-order K-groups ofZ. As introduced before, the Higher algebraic K-groups K n are generally defined as homotopy groups following Quillen’s plus construction; that is, K n (R) := π n (K(R))∀n where K(R) := K 0 (R) × BGL(R) + . Equivalently, we have K n (R) = π n (BGL(R) + )∀n⩾ 1. As we discussed in the remark under Property 6.1.8, higher K-groups don’t have a classical, intuitive expression in general, which makes it hard to obtain a conclusive expression unifying all the K n (Z)∀n. In [28], Lee and Szczarba used (1) in Property 6.1.8 to obtain the following theorem, which takes 29 pages to prove: Theorem6.2.5 ([28]). The groupK 3 (Z) is cyclic of order forty-eight; that is,K 3 (Z) ∼ =Z/48. In order to obtain a decent notion ofK n (Z)∀n in general, we first try generalizing our universe and classify the K-groups of any number field F . If we succeed in classifying all K n (F), then, of course, 86 letting F = Z will directly give all we want. This is a common philosophy in mathematics, which is also what motivated mathematicians to generalize the original Riemann zeta function and examine the corresponding RH in the broader sense, as will be introduced in §6.5.2. Meantime, it provides another motivation and reason of why we should care about K n (F). We will further explore this motivation in §6.5: ClassifyingK n (Z). Prior to that, though, we will go back to some basic algebra knowledge serving as the prerequisite in the next section. 6.3 Algebraprerequisites 6.3.1 GlobalfieldsandDedekinddomains We have talked pretty much aboutanalytic number theory in the first few chapters, so now let’s introduce some important notions in algebraic number theory. Definition6.3.1 (Global fields) . Aglobalfield refers to a field that is either an algebraic number field 3 , namely a finite extension of the field of rational numbers Q, or a function field , namely a finite extension of the extension field k(x) of a finite field k from adjoiningx. In basic algebra, we have the unique factorization theorem (a.k.a., the fundamental theorem of arithmetic) stating that any integer greater than 1 can be represented to be a product of primes, uniquely up to the order of the factors. And then by applying the topological extension model, we de- fine a uniquefactorizationdomain (UFD) as a domain where the unique factorization theorem holds. That is “f(B)=UFD” as mentioned in Example 2.1.6 (as a constitute of an application of ourPatternR). Now imitating this definition, let’s define: Definition6.3.2 (Dedekind domains). ADedekinddomainR is an integral domain where every non- trivial ideal is the product of a finite number of prime ideals. By this definition, it is easy to show some propositions which are equivalent to the previous definition: 4 87 Property6.3.3. SupposeR is an integral domain. Then the following are equivalent: (i)R is Dedekind; (ii) R is integrally closed, Noetherian, and each of its nonzero prime ideals is maximal 5 ; And inR: (iii) every proper ideal is uniquely a product of a finite number of prime ideals; (iv) every nonzero ideal is invertible; (v) every fractional ideal is invertible; (vi) the set of all fractional ideals is a group under multiplication; (vii) every ideal is projective; (viii) every fractional ideal is projective. The proof of this group of properties can be found in [25]. Based on these two notions, we can now induce a conclusive theorem originally introduced by Soule in [43]: Theorem6.3.4. For any dedekind domainR whose field of fractions (a.k.a, fraction field or field of quotients) Frac(R) =: F is a global field, we have: (i)K n (R) ∼ =K n (F)∀n⩾3 odd; (ii) the localization sequence breaks up into exact sequences: 0→K n (R)→K n (F)→ L p K n− 1 (R/p)→ 0∀n⩾2 even. As a direct consequence, sinceZ is Dedekind by definition and Q as its field of fractions is a global field by Definition 6.3.1, let R=:Z, and we will have the following corollary: Corollary6.3.5. K n (Z) ∼ =K n (Q) for anyn⩾3 odd. 88 This provides a crucial fact that we only need to examine K-groups ofQ to help conclude allK n (Z). Meantime, this consequence gives us an inspiration which is to first consider the higher K-groups of some structures related toZ whereby determineK n (Z). We choose this structure to be the next definition. 6.3.2 Ringofintegersandlocalization Definition6.3.6 (Ring of integers). For an algebraic number field (i.e., a finite field extension of Q)E, we habitually denote the integral closureO E/Z ofZ inE (consisting of all elements integral overZ) directly asO E , called theringofintegers inE orringofE-integers. It is easy to prove the following property ofO E : Property6.3.7. LetE be an algebraic number field. (i) The fraction field Frac(O E )=E. (ii)O E is integrally closed. (iii)O E is a noetherian domain. (iv) Every nonzero prime idealI ⊆O E is maximal. Proof. The detailed proof can refer to Chapter 10 in [41]. As a sketch, (i) can be easily proved by the fact that∀α ∈ E ,∃m withmα ∈O E and the definition of fraction field. (ii) follows from (i). For (iii), we can first show O E has an integral basis and thus is a free abelian group of finite rank (under addition). Then any ideal I ⊆ O E is also a finitely generated abelian group (since it’s a subgroup of a finitely generated free abelian group). Thus I is a finitely generated O E -module so it’s a finitely generated ideal. 89 As we can see, the last three items in this property are exactly conditions of (ii) in Property 6.3.3; thus, we conclude: Theorem6.3.8. O E in an algebraic number field E is Dedekind. Next, let’s turn to a similar construction. As we know, the ringQ of rational numbers consists of elements that can be written as a quotient of two integers inZ. HereQ andZ are just two ordinary, particular rings, so naturally, we’d like to generalize this idea to all rings (and correspondingly, modules or abelian groups). And the generalized process, as well as the resulting ring, is called thelocalization 6 , which is to introduce a new ring that consists of all elements of this form m s where m and s are in our existing ring. Localization also has its universal property which can be constructed formally as a definition, but we omit further details in this thesis since they are distanced from our topic. Details can be found in Chapter 10 of [41]. As an example,Z (p) is the localization of the ringZ at the prime idealpZ, whereZ (p) is defined as: Definition6.3.9 (Ring ofp-adic fractions). For each given primep, we set Z (p) :={a/b∈Q:p∤b} i.e. ={a/b∈Q:(p,b)=1} i.e. ={a/b∈Q|a,b / ∈pZ}, and we call this subring ofQ theringofp-adicfractions. Imitating this notation, in general, let’s useG (ℓ) to denote the localization of an abelian groupG at the primeℓ. 6.3.3 TatetwistandBernoullinumbers Recall that thed-thcyclotomicpolynomial is defined as Φ d (x):= Q (x− ζ ) whereζ ranges over all the primitived-th roots of unity (meaning eachd here is the smallest positive integer for whichζ d =1). The 90 name “cyclotomic” originated from a Greek word meaning “circle splitting”. By our translated extension (as introduced in §A.1.2), we may extend this notion to modulus, fields, etc.; for example, we have: Definition 6.3.10. ∀n⩾ 2, a cyclotomic field is defined as E :=Q(ζ n ), whereζ n is a primitiven-th root of unity. Further, we have: Definition6.3.11. For alli∈Z, we define the i-thTatetwist of the cyclotomic moduleµ , writtenµ (i), as the abelian groupµ made into anAut(F)-module by lettingg∈Aut(F) act asζ 7→g i (ζ ). Tate twists are named after John Tate, a doctoral student of Emil Artin, whose research involved many fields of algebraic number theory and arithmetic geometry. About this definition, we have the following property, which proof can be found on Page 515 in [50]. Property6.3.12. SupposeF is algebraically closed. Then ∀i>0,K 2i− 1 ( ¯ F) tors ∼ =µ (i) as anAut(F)-module, whereK 2i− 1 ( ¯ F) tors means the torsion submodule ofK 2i− 1 ( ¯ F). Now suppose F is any field with separable closure ¯ F and Galois group G := Gal( ¯ F/F) ⊆ Aut ¯ F. By this property above, we have the torsion submodule ofK 2i− 1 ( ¯ F), i.e.,K 2i− 1 ( ¯ F) tors , is isomorphic to (can be identified with) the Tate twist as an Aut( ¯ F)-module. Therefore, let’s consider the fixed group µ (i) G def ={a∈µ (i):σ (a)=a∀σ ∈G} and we haveK 2i− 1 ( ¯ F) G tors ∼ =µ (i) G . This means considering the orders of the groupsµ (i) G is of great use to further examine K-groups of separably closed fields (at least to help us determine their orders). Thus, let’s write theorder ofµ (i) G asw i (F) ifµ (i) is a finite (thus cyclic) group. In order words, we now can writeµ (i) G∼ =Z/w i (F). 91 In particular, when the field F =:Q, we will henceforth writew i (Q) directly asw i . As shown in Ex- ample 2.1.2 in [50], we havew i (Q( √ − 1))=4 andw i =2∀i odd. Fori even, we havew i =w i (Q( √ − 1)), andℓ|w i exactly when(ℓ− 1)|i . Moreover, we have the following theorem: Theorem6.3.13 ([50]). Ifi=2k is even, thenw i is exactly the denominator of f B k 4k . This theorem provides an easy way to compute all the exact values ofw i . The following is the first six values: w 2 =24=2 3 · 3 w 4 =240=2 4 · 3· 5 w 6 =504=2 3 · 3 2 · 7 w 8 =480=2 5 · 3· 5 w 10 =1320=2 3 · 3· 5· 11 w 12 =65,520=2 4 · 3 2 · 5· 7· 13 ... ... Note that in this chapter, we shall always take our second definition (Definition 4.3 ′ ) of Bernoulli numbers, i.e., f B k . The reason is, as said in §4.2, this second definition is more popularly used in many algebra and topology fields. The advantage of this choice here is that all of f B k are positive since they are not signed; hence it’ll be conducive to representing the order of K n (Z). Moreover, we denote c k /d k as the reduced expression for f B k /4k. That is,            c k :=Nu f B k /4k =Nu f B k d k :=De f B k /4k =De f B k · 4k whereNu(p) andDe(p) represent the numerator and denominator, respectively, of the reduced form for a fractionp in general. Then this Theorem 6.3.13 tells usd k is just equal tow 2k . So we have f B k /4k =c k /w 2k . Similarly, for a given primeℓ, we may writew (ℓ) i (F) for the order ofµ (ℓ) (i) G , where the latter is the localization as introduced in the previous section. 92 6.4 Étalecohomology 6.4.1 Fromsingularhomologytoétalecohomology In algebraic topology, we usually denote the set of all singular n-simplices in X by S n (X). Then we write the freeR-module generated by the setS n (X) asC n (X), meaningC n (X) consists of all such maps c : S n (X) → R so that c(σ ) = 0 for all but finitely many elements σ inS n (X). Finally, define the boundary homomorphism∂ n : C n (X)→ C n− 1 (X) by∂ n (σ ) := P n i=0 (− 1) i σ ◦ F i . Based on them, we have: Definition 6.4.1 (homology modules and homology groups). The n-th homology module of X with coefficients in R is defined as H n (X;R):=Z n (X)/B n (X). HereZ n (X):={c∈C n (X);∂ n c=0} is thekernel of the boundary map∂ n while the “divisor”B n (X)= {c∈C n (X);∃c ′ ∈C n+1 (X),c=∂ n+1 c ′ } is theimage of∂ n+1 . We usually omit the coefficient ring R and write it as H n (X) in the context. In particular, when R =:Z, aZ-module is an abelian group and thus H n (X;Z) are thehomologygroups ofX. Notice we can easily prove∂ n− 1 ◦ ∂ n = 0∀n, which impliesB n (x)⊆ Z n (x)⊆ C n (x). That means homology modules are well-defined. And we have: ··· ∂ n+2 −→ C n+1 (X;R) ∂ n+1 −→ C n (X;R) ∂n −→ C n− 1 (X;R) ∂ n− 1 −→ ... For example, when X = ∅, we haveS n (X) = {σ :∆ n →∅} = ∅, so C n (X) = {free R-module generated by the empty set}=0. ThusH n (∅)=0∀n. Definition 6.4.2. A singular n-cochain (valued inR) is a map c : S n (X) → R. We denote the set of singular cochainsc (valued inR) asC n (X) which clearly is anR-module. 93 Moreover, we have Definition6.4.3. Thecoboundary of a cochainc∈C n (X;R) is the cochaindc∈C n+1 (X;R) defined bydc(σ ):= P n+1 i=0 (− 1) i c(σ ◦ F i )∈R∀(n+1)-simplexσ . So we now can write the following sequence of vector spaces and linear maps, where mapsd n are in the opposite directions of the boundary maps∂ n : ··· d n− 2 −→ C n− 1 (X;R) d n− 1 −→ C n (X;R) dn −→ C n+1 (X;R) d n+1 −→ ... Similar to Definition 6.4.1 above, we may define: Definition6.4.4. Then-thcohomologyspace ofX with coefficients in R is defined as the vector space H n (X;R):=Z n (X;R)/B n (X;R) HereZ n (X;R) is the kernel of the coboundary mapd n : C n (X;R) → C n+1 (X;R) whileB n (X;R) is the image of the previous mapd n+1 :C n− 1 (X;R)→C n (X;R). We find that Definition 6.4.1 and Definition 6.4.4 are comparable except we changed the boundary maps∂ n to coboundary mapsd n , so the naming of cohomology spaces is also obtained in a natural way. Likewise, by definition, it is easy to verify d◦ d = 0, meaning the kernelZ n (X;R) indeed contains the imageB n (X;R). Next, similar to our singular cohomologies, we turn to the definition of the étale cohomology. First, let’s review tangent spaces. As we said in §2.1.1, the topological extension model is crucial in generalizing math. When discussing derivatives, we know the derivation operator satisfies the product (Leibniz) rule, so let’s do a topological extension on this rule to obtain: 94 Definition 6.4.5 (Derivations and tangent spaces). Let M be a smooth manifold p ∈ M. A linear map v :C ∞ (M)→R is called aderivation atp if it satisfies v(fg)=f(p)vg+g(p)vf ∀f,g∈C ∞ (M). The set of all derivations ofC ∞ (M) atp is denoted byT p M, called thetangentspace toM at pointp. An element ofT p M is called atangentvector atp. As in the case of open sets, this extension enables us to consider all parallel notions in differentiation. For example, we have: Definition6.4.6 (Differentials) . SupposeM andN are smooth manifolds andF : M → N is a smooth map.∀p∈M, we define a map dF p :T p M →T F(p) N, known as thedifferential ofF atp. We may easily verify some properties of differentials above parallel to our original definition, such as linearity, chain rule, etc. (See Page 55 in [27].) Here let’s highlight one property: Property6.4.7 ([27]). IfF is a diffeomorphism, then dF p :T p M →T F(p) N is an isomorphism, and(dF p ) − 1 =d F − 1 F(p) . In this property, if we give the conclusion “dF p is an isomorphism” a name, then it will directly lead to the notion of étale maps: Definition6.4.8 (Étale maps). SupposeM andN are two differentiable manifolds. A C ∞ mapφ:N → M is said to beétale at pointn∈N if the map on tangent spacesdφ:T n N →T φ(n) M is an isomorphism. By our translated extension model, we can immediately deduce the following definitions for algebraic and smooth varieties: 95 Definition6.4.9. SupposeX andY are nonsingular algebraic varieties over an algebraically closed field k. We call a regular mapφ:Y →X étale at pointy∈Y ifdφ:T y Y →T φ(y) X is an isomorphism. Definition6.4.10. LetX andY be smooth varieties over an algebraically closed field k. Then a regular mapφ:Y →X is said to beétale if it is étale at all pointsy∈Y . Recall that the(affine)variety defined by F ⊆ k[X] (where[X]:=[x 1 ,...,x n ]) is Var(F):={a∈k n :f(a)=0∀f(X)∈F}. Indeed, the name variety arose from the translation in 1869 of the German word Mannigfaltigkeit (mod- ernly translated as “manifold”) used by Riemann. In modern language, we usually call the “affine variety” scheme. With all these étale definitions, we can now turn to the construction of its topology; that is, to define the open sets and corresponding definitions of continuity, neighborhood, etc. For the étale topology on a varietyX, theopensets can be defined as the étale morphisms U → X. Note that a family of étale morphisms(U i →U) i∈I overX is a covering ofU ifU = S φ i (U i ). The details can be found in Section 4 of [33]. Additionally, anétaleneighborhood of a pointx∈X is an étale map U →X (as defined above) with a point u∈U mapping tox. With this topology, we will define “sheaves” which is central to the definition of étale cohomology groups. So let’s first introduce the following definition: Definition6.4.11. We denoteEt/X as thecategory whoseobjects are the étale mapsU →X and whose arrows are the commutative diagrams: V U X i j◦ i j 96 where the mapsj◦ i:V →X andj :U →X are étale (soi:V →U is also étale by definition). Recall that a contravariant functor T : C → D (whereC andD are categories) is a function such that the following hold: (See Chapter 6 in [41].) (i) ifC∈obj(C), thenT(C)∈obj(D); (ii) iff :C→C ′ inC, thenT(f):T (C ′ )→T(C) inD; (iii) ifC f −→ C ′ g −→ C ′′ inC, thenT (C ′′ ) T(g) −→ T (C ′ ) T(f) −→ T(C) inD andT(gf)=T(f)T(g); (iv)∀A∈obj(C),T (1 A )=1 T(A) . Definition6.4.12. We say a contravariant functorF :Et/X →Ab is apresheaf for the étale topology onX. Furthermore, this functor is asheaf if the sequence F(U)→ Y i∈I F(U i )⇒ Y (i,j)∈I× I F(U i × U U j ) is exact for all étale coverings(U i →U). Finally, we can turn to the definition of the étale cohomology groups. By Theorem 11.34 in [31], the category of sheavesAb(C) is an abelian category with enough injectives. Hence by the derived functors ofF 7→ F(X) we may define étale cohomology groups as in the classical case. To put it strictly, here comes our final definition: Definition6.4.13 (Étale cohomology of sheaves). LetS be a scheme andF be an abelian sheaf. We define thei-th étale cohomology group ofF overS as H i et (S,F):=R i Γ( S,F), that is, thei-th right derived functor of the global sections functorΓ: F 7→H 0 (S,F). 97 6.4.2 LocalzetafunctionandWeilconjectures Let’s first introduce another kind of generalization of the original zeta function. In 1923 [3], Emil Artin first studied the following function: Definition6.4.14 (Local zeta function). We define Z(V,s) := exp P ∞ m=1 Nm m (q − s ) m , which is a func- tion on a non-singular projective algebraic varietyV over the finite field F q (withq elements), whereN m is the number of points ofV defined over the degree- m extensionF q m of the finite field F q . We callZ(V,s) thelocalzetafunction. Sometimes we write the aboveZ(V,s) asZ V (s), as used in Section 9.12 of [45]. Following the original RH, Artin put forward another hypothesis with respect to the local zeta function, called “the Riemann hypothesis for curves over finite fields”. Unlike the sustaining mystery of RH, this “translated” hypothesis was already proved by André Weil in the 1940s. This further motivated him to conceive the Weil conjec- tures, including a further extended analogue of RH on algebraic varieties over finite fields which has also been proved by Pierre Deligne [9]. Theorem6.4.15 (Weil conjectures). LetX be a smooth, geometrically connected,n-dimensional projective algebraic variety, defined over a finite field k =F q . Forsimplicity,weomitthevarietyX inthelocalzetafunctionZ(X,s)andwriteitasZ(s)in the following statements: (i) (Rationality)Z(s) is a rational function ofs, that is, Z(s) lies in the fraction field Frac (k[s]) =: Z[s] of k[s]. (ii) (Functional equation). The local zeta function satisfies Z 1 q n s = ± q nE/2 s E Z(s), where E = ∆ 2 is the self-intersection of the diagonal ∆ ,→ X× X. Equivalently, under the same condition, ζ (n− s) = 98 ± q nE 2 − Es ζ (s). (iii) (Analogue of Riemann hypothesis). There are polynomials                      P 0 (s)=1− s P i (s)= Q j (1− α i,j s)∈Z[s] fori⩽2n− 1 P 2n (s)=1− q n s whereα i,j arealgebraicintegerswith|α i,j |=q i 2 ,suchthatZ(s)= P 1 (s)·P 3 (s)··· P 2n− 1 (s) P 0 (s)·P 2 (s)··· P 2n (s) . TherootsofP i (s), fori=0,...,2n, have absolute valueq − i 2 . (iv)(Bettinumbers,[34])Assumingthevalidnessof(iii),wedefinethe“ i-thBettinumber ofX”asb i (X):= deg(P i (s)). We have the following two statements: (1)E = P 2n i=0 (− 1) i b i (X). (2) LetR be a finitely generated Z-subalgebra of the field C of complex numbers, e X be a smooth projective schemeoverSpecR,andP ∈SpecRbeaprimeidealsuchthatR/P =F q and e X× SpecR SpecR/P =X hold. Then we have b i (X)=dim Q H i ( ˜ X× SpecR SpecC) an ,Q . Statement (ii) here is an analogue of the functional equations of the Riemann and Hurwitz zeta func- tions proven earlier in this thesis. Statement (iii) is also about the distribution of roots of some zeta func- tion, and that’s why we call it the Riemann hypothesis—as we see, this naming is also obtained by the topological extension model, just like the naming of open sets in a topological space. Historically speaking, Statement (i) “Rationality” was proven by Dwork in 1960. Then two years later, Lubkin proved all these conjectures other than (iii) “RH” under certain restrictions, but his proof referred to some other conjectures which hadn’t been proven (see [45]). In 1965, Grothendieck proved (ii) “Functional equation”. And finally in 1974, as mentioned above, the RH conjecture was completely proved by Deligne. 99 Interestingly, we are able to prove an analogue of RH under certain restrictions, but unfortunately, the same or analogous method to solve the “translated” RH could not be translated into solving the original RH, which remains unsolved up till today. Even so, completing the proof of this RH in Weil conjectures can give us some inspiration or at least some confidence to conquer the mystery of the original RH, and this is where the significance of a great conjecture lies. 6.5 ClassifyingK n (Z) At this point, we will utilize all of the definitions and properties introduced in our last section to build the connection ofK n (Z). We supposer 1 ,r 2 are the numbers of real and non-conjugate complex imbeddings σ ofF (intoR andC), resp.; that is,            r 1 =#{σ :F ,→R} r 2 = 1 2 #{σ :F ,→C,σ ̸= ¯σ } . And in general we let|X| (also written as#X) denote the cardinality of any finite set X. 6.5.1 K n (Z)foroddn As defined above, we suppose O S is a ring of S-integers in a number field F , and let R =: O S [1/ℓ] for some primeℓ given. And as mentioned in the last part of §6.2, our next goal is to determineK n (O S ) or K n (R) and build a connection between it andK n (Z). First by Theorem 6.3.4 we see K n (O S ) Th6.3.4 ∼ = K n (Frac(O S ))=K n (R) since Theorem 6.3.8 tells usO S is a Dedekind domain. Hence the ring of integersO S andR =O S [1/ℓ] actually have the same higher K-groups for each primeℓ. That is, considering the localization sequence in Theorem 6.3.4 and their localizations ofℓ, we can get K n (O S ) (ℓ) =K n (R) (ℓ) . 100 Moreover, we have: Theorem6.5.1 ([51]). ∀n⩾2, for some primeℓ given, K n (O S ) (ℓ) ∼ =K n (R) (ℓ) ∼ =                H 2 et (R;Z ℓ (i+1)) forn=2i>0; Z r 2 (ℓ) ⊕ Z/w (ℓ) i (F) forn=2i− 1,i even; Z r 2 +r 1 (ℓ) ⊕ Z/w (ℓ) i (F) forn=2i− 1,i odd. Proof. The first isomorphism has been shown above the theorem. A proof of the second part can be found in Chapter 8 in [50]. This theorem specifically computes all the odd torsion subgroups of K n (O S ) as shown in the last two lines of the bracket. Now since K n (O S ) ∼ = K n (F) (by Theorem 6.3.4), our goal is equivalent to classifying K n (O S ). and correspondingly, examining the torsion subgroup of K n (O S ). By Theorem 6.5.1 above, we have found the odd torsion K n (O S ) (ℓ) ∀n = 2i− 1. Hence next we have to classify the 2-primary tor- sion. By definition, since K n (O S ) is finite, the 2-primary subgroup ofK n (O S ) isK n (O S ;Z/2 ∞ )φ def = π n (K(O S );Z/2 ∞ )∀n ∈Z, namely the mod 2 ∞ algebraic K-groups ofO S . The detailed discussion can be found in [39]. Moreover, we can deduce: Theorem6.5.2 ([39]). LetF be a real number field with R:=O F 1 2 denoting its ring of2-integers, and writeW(i) for the union of the étale sheavesZ/2 ν (i) in general. Then the mod2 ∞ algebraic K-groups ofR are as follows: 101 K n (R;Z/2 ∞ ) ∼ =                                                          H 0 (R;W(4k)) ∼ =Z/w 4k (F) forn=8k, H 1 (R;W(4k+1)) forn=8k+1, H 0 (R;W(4k+1)) ∼ =Z/2 forn=8k+2, H 1 (R;W(4k+2)) forn=8k+3, (Z/2) r 1 ⋊H 0 (R;W(4k+2)) forn=8k+4, (Z/2) r 1 − 1 ⋊H 1 (R;W(4k+3)) forn=8k+5, 0 forn=8k+6, ˜ H 1 (R;W(4k+4)) forn=8k+7. Remark: In this theoremH 0 (R;W(4k+2)) ∼ =Z/w 4k+2 (F), and the extension in degreen=8k+4 is abstractly isomorphic to(Z/2) r 1 − 1 ⊕ Z/2w 4k+2 (F). And in general,A⋊B represents an abelian group extension of B by A, i.e., an abelian group containing A with quotient B; for example, the extension in n=8k+5 means some abelian group extension ofH 1 (R;W(4k+3)) by(Z/2) r 1 − 1 . This theorem can be easily extended toR :=O S that is a ring ofS-integers inF containingO F [ 1 2 ]; see Section VI.9 in [50] for details. So now we can immediately obtain: Theorem6.5.3. K n (O S ;Z/2 ∞ ) ∼ =                                                          Z/w 4k (F) forn=8k, H 1 (O S ;Z/2 ∞ (4k+1)) forn=8k+1, Z/2 forn=8k+2, H 1 (O S ;Z/2 ∞ (4k+2)) forn=8k+3, Z/2w 4k+2 ⊕ (Z/2) r 1 − 1 forn=8k+4, (Z/2) r 1 − 1 ⋊H 1 (O S ;Z/2 ∞ (4k+3)) forn=8k+5, 0 forn=8k+6, e H 1 (O S ;Z/2 ∞ (4k+4)) forn=8k+7. Thus now by this Theorem 6.5.3 and Theorem 6.5.1, we have concluded: 102 Theorem6.5.4. If a number field F hasr 1 >0 real embeddings andi:= n+1 2 , then K n (F) ∼ =                        Z r 1 +r 2 ⊕ Z/w i (F), n≡ 1 (mod 8), Z r 2 ⊕ Z/2w i (F)⊕ (Z/2) r 1 − 1 , n≡ 3 (mod 8), Z r 1 +r 2 ⊕ Z/ 1 2 w i (F), n≡ 5 (mod 8), Z r 2 ⊕ Z/w i (F), n≡ 7 (mod 8). (6.1) The detailed proof can be found in either [39] or [50]. In particular, again, we want to computeK n (Z) by this theorem. So By Corollary 6.3.5, it suffices to compute K n (Q) (∀n⩾ 3 odd). Thus lettingF =:Q we have            r 1 =1 r 2 =0 by definition, so (6.1) becomes K n (Z) ∼ = K n (Q) ∼ =                        Z⊕ Z/w i (Q), n≡ 1 (mod 8), 1⊕ Z/2w i (Q)⊕ 1, n≡ 3 (mod 8), Z⊕ Z/ 1 2 w i (Q), n≡ 5 (mod 8), 1⊕ Z/w i (Q), n≡ 7 (mod 8) where each i = n+1 2 and 1 here represents the trivial ring. That is, K n (Z) ∼ =                        Z⊕ Z/w 4k+1 , n=8k+1, Z/2w 4k+2 , n=8k+3, Z⊕ Z/ 1 2 w 4k+3 , n=8k+5, Z/w 4k+4 , n=8k+7. (6.2) 103 Recall thatw i is short forw i (Q), andw 4k+1 =w 4k+3 =2 sincew ∀odd =2 as mentioned in §6.3.3. By Theorem 6.3.13,w 4k+2 =w 4k+4 =d 2k+1 in the second and last lines of the above (6.2). Note thatZ/n in general is short for the cyclic groupZ/nZ. That is to say, now we have K n (Z) ∼ =                        Z⊕ Z/2, n=8k+1, Z/2w 4k+2 , n=8k+3, Z, n=8k+5, Z/w 4k+4 , n=8k+7. Meanwhile, from it we can read off the order of K n (Z) forn odd: |K n (Z)|=                        ∞, n=8k+1, 2w 4k+2 , n=8k+3, ∞, n=8k+5, w 4k+4 , n=8k+7. Extracting the second and last (finite) rows, we obtain |K n (Z)|=        2w 2k , n=4k− 1 fork odd, w 2k , n=4k− 1 fork even. (6.3) i.e., |K 4k− 1 (Z)|=        2w 2k , k odd, w 2k , k even. (6.4) 6.5.2 DedekindzetafunctionandK n (Z)forevenn So what about other values ofn? Can we deduceK n (Z) forn even from results for oddn above? Let’s further explore in this subsection. 104 Remember our main topic of this thesis is still zeta functions, and the next theorem we are about to introduce builds a bridge between zeta functions and K-theory, and it will help us clarifyK n (Z) for evenn. But before giving this theorem, let’s introduce another kind of generalization of the Riemann zeta function. Definition6.5.5 (Dedekind zeta function). We define the Dedekindzetafunction of field K as ζ F (s):= X I⊆O F (N F/Q (I)) − s , where F is an (algebraic) number field, O F denotes the ring of integers of F , P I⊆O F means I ranges through all non-zero ideals ofO F , andN F/Q (I) denotes the absolute norm ofI (equal to the index[O F :I], i.e., the cardinality of quotient ringO F /I, also denoted#(O F /I)). Based on this function, one can also obtain many previously introduced concepts analogous toζ (s), such as analytic continuation, Euler product formula, and the corresponding RH − called the Extended Riemann Hypothesis (ERH). ERH asserts that all nontrivial zeros ofζ F (s) are on the lineℜ(s) = 1/2 in C. Notice unlike the local zeta function (Definition 6.4.14), this Dedekind zeta function includes ζ (s) as its special case (whenF =Q), meaning proving this ERH will automatically verify the original RH. In 1990 [53], Andrew Wiles, the mathematician who solved Fermat’s last theorem, proved the following theorem about our Dedekind zeta function and étale cohomology groups ofO S : Theorem6.5.6. SupposeF is a totally real number field (i.e., r 2 = 0),ℓ is odd, andO S :=O F [1/ℓ].∀ positive even integers 2k, there exists a rationalu k ∈Q , prime toℓ, such that ζ F (1− 2k)=u k H 2 et (O S ,Z ℓ (2k)) H 1 et (O S ,Z ℓ (2k)) . Note that ifF is not totally real, thenζ F (s) has a pole of orderr 2 ats=1− 2k. 105 Also, we have the numerator and denominator on the right-hand side of the foregoing equation are both finite (as shown in Ex. 8.1 and 8.2 in [50]). Prior to the introduction of Theorem 6.5.6, in 1973, Lichtenbaum had made a conjecture in [30] about the values ofζ F (s) at negative odd integers: Theorem6.5.7. Suppose field F is totally real (i.e.,r 2 =0) andGal(F/Q) is abelian. We have∀k⩾1, ζ F (1− 2k)=(− 1) kr 1 2 r 1 |K 4k− 2 (O F )| |K 4k− 1 (O F )| . (6.5) Now by Theorem 6.5.6 above plus Theorem 6.3.4 as we introduced in §6.3.1, we can get a proof of this conjecture (see [50]), so this can now be called a “theorem”. In particular, when settingF =:Q (which is totally real and whereGal(Q/Q) is trivially abelian), we haveO Q =Z and            r 1 =1 r 2 =0 , so the equation (6.5) in Theorem 6.5.7 becomes ζ (1− 2k)=2(− 1) k |K 4k− 2 (Z)| |K 4k− 1 (Z)| . (6.6) As mentioned above, these two theorems are essential to this chapter of the thesis. By (4.6) in §4.3 and (4.4) in §4.2,∀k ⩾ 1, we have the following formula under the unsigned f B k definition of Bernoulli numbers ζ (1− 2k)=(− 1) k f B k 2k ; (4.6 ′ ) equivalently, f B k 4k = (− 1) k 2 ζ (1− 2k), (6.7) 106 which builds a relation between values ofζ (s) and Bernoulli numbers f B n . So now we get relation between K n (Z) groups and f B n : |K 4k− 2 (Z)| |K 4k− 1 (Z)| = f B k 4k . By plugging Equation (6.4) into this equation above, we immediately get |K 4k− 2 (Z)|=        f B k 2k w 2k , k odd, f B k 4k w 2k , k even. (6.8) By the remark under Theorem 6.3.13, we substitute f B k /4k =c k /w 2k , i.e., f B k =4k c k w 2k , into this equation above, we find the two w 2k cancel, so we now have: |K 4k− 2 (Z)|=        2c k , k odd, c k , k even. (6.9) Equivalently, |K n (Z)|=        2c 2k+1 ,n=8k+2, c 2k+1 ,n=8k+6. (6.10) 6.5.3 Finalconclusion In conclusion, now we have found the following result: Theorem6.5.8. ∀n̸≡ 0(mod 4),n>1, we have: (1)n=8k+1⇒K n (Z) ∼ =K n (Q) ∼ =Z⊕ Z/2; (2)n=8k+2⇒|K n (Z)|=2c 2k+1 ; (3)n=8k+3⇒K n (Z) ∼ =K n (Q) ∼ =Z/2d 2k+1 ; (4)n=8k+5⇒K n (Z) ∼ =K n (Q) ∼ =Z; 107 (5)n=8k+6⇒|K n (Z)|=c 2k+1 ; (6)n=8k+7⇒K n (Z) ∼ =K n (Q) ∼ =Z/d 2k+1 . Abovek isanonnegativeinteger,andc k ,d k arethenumeratoranddenominatorofthereducedexpressionfor the Bernoulli numbers f B k over4k. Examining our results so far, one may ask: what about the other two cases of n = 8k +4 and n = 8k+8 (that is, the case forn≡ 0(mod 4),n > 1)? Notice the results about the order ofK 8k+2 (Z) and K 8k+6 (Z) come fromK 4k− 2 (Z) in (6.9) deduced from Theorem 6.5.7 and (6.6); but (6.6) cannot give us the corresponding results forK 8k+4 (Z) orK 8k+8 (Z), so this whole method seems to have failed. Besides, in the previous theorem we only gave theorder ofK n (Z) forn even but didn’t even give the explicit formula for them. In fact, in 2000 [38], Rognes provedK 4 (Z) is trivial. For other multiples of4, denoted4k(k⩾2), we only have known the orders ofK 4k (Z) are products of irregular primesℓ withℓ>10 8 . Also, mathemati- cians have conjecturedK 8k+4 (Z) andK 8k+8 (Z) all equal0 actually, which is (equivalent to) the Vandiver’s Conjecture (see [50]). Moreover, given this conjecture is true, we also will obtainK 8k+2 (Z)=Z/2c k and K 8k+6 (Z) = Z/c k . That is to say, once we prove the validness of (any equivalent form of) Vandiver’s Conjecture, all K-groups ofZ will be completely classified. In summary, now K-theorists have concluded the following table [50] with 8 rows, where(0)? means some finite group unknown but conjectured 0: K 0 (Z)=Z K 8 (Z)=(0?) K 16 (Z)=(0?) K 8k (Z)=(0?),k⩾1 K 1 (Z)=Z/2 K 9 (Z)=Z⊕ Z/2 K 17 (Z)=Z⊕ Z/2 K 8k+1 (Z)=Z⊕ Z/2 K 2 (Z)=Z/2 K 10 (Z)=Z/2 K 18 (Z)=Z/2 K 8k+2 (Z)=Z/2c 2k+1 K 3 (Z)=Z/48 K 11 (Z)=Z/1008 K 19 (Z)=Z/528 K 8k+3 (Z)=Z/2d 2k+1 K 4 (Z)=0 K 12 (Z)=(0?) K 20 (Z)=(0?) K 8k+4 (Z)=(0?) K 5 (Z)=Z K 13 (Z)=Z K 21 (Z)=Z K 8k+5 (Z)=Z K 6 (Z)=0 K 14 (Z)=0 K 22 (Z)=Z/691 K 8k+6 (Z)=Z/c 2k+2 K 7 (Z)=Z/240 K 15 (Z)=Z/480 K 23 (Z)=Z/65520 K 8k+7 (Z)=Z/d 2k+1 Table 6.1: K-groups ofZ 108 By direct substitution, we can now compute many special values of the Riemann zeta function by Equation (6.6): ζ (1− 2k) = 2(− 1) k |K 4k− 2 (Z)| |K 4k− 1 (Z)| (deduced from Theorem 6.5.7), in the algebraic-K-theory perspective. k =:1⇒ζ (− 1)=2(− 1) 1 |K 2 (Z)| |K 3 (Z)| =− 2× 2 48 =− 1 12 ; k =:2⇒ζ (− 3)=2(− 1) 2 |K 6 (Z)| |K 7 (Z)| =2× 1 240 = 1 120 ; k =:3⇒ζ (− 5)=2(− 1) 3 |K 10 (Z)| |K 11 (Z)| =− 2× 2 1008 =− 1 252 ; k =:4⇒ζ (− 7)=2(− 1) 4 |K 14 (Z)| |K 15 (Z)| =2× 1 480 = 1 240 ; k =:5⇒ζ (− 9)=2(− 1) 5 |K 18 (Z)| |K 19 (Z)| =− 2× 2 528 =− 1 132 ; k =:6⇒ζ (− 11)=2(− 1) 6 |K 22 (Z)| |K 23 (Z)| =2× 691 65520 = 691 32760 ; ... We see all of them do agree with our previous results about values ofζ (s) at negative odd numbers in §4.3. Moreover, providing Theorem 6.5.8 holds, we find a new way to verify (6.6) and (6.7) by the trivial computations (where f B k are still as defined in §4.2): |K 4k− 2 (Z)| |K 4k− 1 (Z)| Th6.5.8 === c k d k def = f B k 4k by(6.7) === (− 1) k 2 ζ (1− 2k)∀k⩾1. 6.6 Futureresearch As the final section of this chapter, we present some additional ideas and hypotheses that will require further research. 109 In the previous Property 6.1.8, (1) is significant, because we have K 1 (R) ∼ = H 1 (GL(R),Z),K 2 (R) ∼ = H 2 ([GL(R),GL(R)],Z), and now K 3 (R) ∼ = H 3 (St(R),Z), in which a sequence consisting of closely related spaces has been formed. From the sequence, it is natural to ask whether we may extend this sequence and define K i directly usingH i instead ofπ i . The author tried to find that pattern but he didn’t, and he has not yet found any reference about this—such direct connections as this between K i and H i stop ati=3 in all references that he’s examined. 7 Moreover, as we know mathematicians usually are not satisfied with extending a notation to merely natural numbers, but they want to include all real, even all complex, numbers and hopefully to unify some math notions in seemingly different fields by them. Indeed, negative K-groups K − n (R) are already produced by Bass when researching the fundamental theorem of algebraic K-theory. It is defined as K − n (R) := coker(K − n+1 (R[t])⊕ K − n+1 R t − 1 → K − n+1 R t,t − 1 ). But we are not go- ing to further examine it in this thesis, since it is not significantly related to values of ζ (s). First, by K n (R) = π n (BGL(R) + )∀n ⩾ 1, why don’t we do the topological extension from n ∈ N + to n ∈ Z whereby define the negative homotopy groups π − 1 ,π − 2 ...? When the author was thinking of that, he found mathematicians have done similar constructions (as introduced in [32]), but he thinks they can be further refined. Second, based on his knowledge, the author hasn’t found any definitions as to n = i or n=3/4 forK n (R). He is convinced that a natural extension ton∈C can and will also be built similarly. If that can be achieved, algebraic K-theory will be made continuous, which makes it possible to define differentials or integration accordingly. Maybe a new math subject called Differential K-theory will be naturally produced through that. 8 Third, the author conjectures a connection between K − n (Z) and ζ (2k− 1) (values at positive odd integers) can naturally be built, provided that this definition of negative K-theory is indeed a natural, “analytic” continuation of higher algebraic K-groups. Lastly, as we see, the case ofn≡ 0(mod 4),n > 1, which is not included in Theorem 6.5.8, is much harder, in the same way determining the values of ζ (s) at positive odd numbers is much harder than at 110 other integers. In addition, it is the connection between Bernoulli numbers and values ofζ (s) at positive integers that let us conclude the case forn=8k+2 and8k+6. Therefore, these observations also lead us to conjecture that we can deduce a formula forK 8k+4 (Z) from the formula forζ (2k+1) in a direct way, and vice versa; namely once we prove Vandiver’s conjecture or similar results, we may be able to deduce the mysteriousζ (2k+1) from them. 111 Chapter7 Conclusions In this thesis, we mainly examined the Riemann zeta function and its application to algebraic K-theory. Specifically, in Chapter 1, we introduced the Riemannzetafunction via basicp-series and theprimecount- ing function related to the distribution of primes. Then by considering the prime zeta function, we further deduced the product form of Euler’s product formula. In Chapters 2 and 3, we introduced Hurwitz’s zeta function, analytically extended ζ (s) to all the complex plane (except for the point s = 1), and discussed Hurwitz’s functional equation (which is the corresponding generalized form of the Riemann’s functional equation). We first proved Riemann’s functional equation using Hurwitz’s formula in Chapter 3, and in the proof of Hurwitz’s formula, we introduced theperiodiczetafunction which also includes the Riemann zeta function as a special case. At the end of Chapter 3, we provided an elementary proof of Riemann’s functional equation by investigating the Jacobi theta function. Starting with the Basel problem and an in- troduction to Bernoulli polynomials, Chapters 4 and 5 focus on special values of zeta functions. Zeros of ζ (s), which are related to the famous unsolved Riemann hypothesis, are especially studied in Chapter 5, including an introduction of many modern results in this area. Chapter 6 further generalized the Reimann zeta function to the local zeta function and Dedekind zeta function, each of which corresponds to an ana- logue of the Riemann hypothesis. In order to fully understand notions in K-theory and the connection between K n (Z) and ζ (s), we presented a few pieces of algebra knowledge, such as Tate twist and étale 112 cohomology. We also put into Appendix A some prerequisite knowledge about the groups K 1 , K 2 , and K 3 . Additionally, the Riemann zeta function can be extended to a Dirichlet series, called theDirichletL- function, in another way according to the idea presented in Chapter 3 above; in this case, the RH becomes the generalized Riemann hypothesis (GRH). The Euler product formula deduced in this thesis can also be generalized to hold for all multiplicative arithmetical functions. But as we see in Chapter 6, in fact, mathematicians chose to generalize the Riemann zeta function to the Dedekind (instead of Dirichlet) zeta function when establishing the connection between zeta functions and K-groups ofZ. That’s why we didn’t introduce more about the Dirichlet L-function in this thesis. Can we build another connection between the Dirichlet L-function and K-groups of some other rings? These questions are still to be explored. Lastly, let’s add another comment on the famous hypothesis RH related to the distribution of zeros of ζ (s). RH has not yet been found to have any counterexamples, even after verifying hundreds and millions of zeros, which probably indicates that RH is true. Yet, Littlewood and his students once demonstrated that if RH holds, the critical point at which the sign ofLi(x)− π (x) changes from positive to negative for the first time will occur somewhere before e e e 79 (which is an enormous number), while all the evidence prior to that time had suggested thatLi(x)− π (x)> 0 always held. This implies that the validity of RH does not have to, and can never, be verified by enumerating zeros one by one; rather, what is required is an authentic mathematical proof. In the process of proving a proposition, we usually gain more discoveries than the proposition itself. For example, investigating the proof of Fermat’s last theorem has led to the development of seven independent mathematical disciplines, including group theory. Therefore, even though completing the proof of RH may still be a long journey, many unexpected results are likely to be attained in the process, and their combination constitutes a “positive definite” driving force that stimulates the growth of mathematics and scientific research, and will never be altered by “nonsingular” mistakes. 113 Such development means that mankind will certainly soon unravel all the mysteries of the Riemann zeta function, including its explicit pattern of values at positive odd integers, the relationship between K- groups, and even the Riemann hypothesis. Finally, we conclude with a quotation from Hilbert’s Konigsberg speech, which has also been carved on his tombstone in Göttingen—“We must know, we shall know!” 114 Bibliography [1] L.V. Ahlfors and Karreman Mathematics Research Collection. Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable. International series in pure and applied mathematics. 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Zeta-functions: An Introduction to Algebraic Geometry. Research notes in mathematics. Pitman, 1977.isbn: 9780273010388.url: https://books.google.com/books?id=Rb8PAQAAMAAJ. [46] L.S.P.G.E.C. Titchmarsh, E.C. Titchmarsh, A. Wolffe, D.R. Heath-Brown, Oxford University Press, and E.C.T. Titchmarsh. The Theory of the Riemann Zeta-function. Oxford science publications. Clarendon Press, 1986.isbn: 9780198533696.url: https://books.google.com/books?id=1CyfApMt8JYC. [47] Richard Tolimieri and Myoung An. “Poisson summation formula”. In: Time-Frequency Representations. Springer, 1998, pp. 47–56. [48] C de la Vallee-Poussin. “Recherches analytiques sur la theorie des nombres premiers. Deuxieme partie: Les fonctions de Dirichlet et les nombres premiers de la forme lineaire Mx+ N”. In:Ann.Soc. Sci. Bruxelles 20 (1896), 281 362 (1896). [49] Alfred Van der Poorten and R Apéry. “A proof that Euler missed...” In: The Mathematical Intelligencer 1.4 (1979), pp. 195–203. [50] C.A. Weibel. TheK-book: An Introduction to AlgebraicK-theory. Graduate Studies in Mathematics. American Mathematical Society, 2013.isbn: 9780821891322.url: https://books.google.com/books?id=Ja8xAAAAQBAJ. [51] Charles Weibel. “Algebraic K-theory of rings of integers in local and global fields”. In: Handbook of K-theory 1 (2005), p. 2. [52] John HC Whitehead. “Simple homotopy types”. In: American Journal of Mathematics 72.1 (1950), pp. 1–57. [53] Andrew Wiles. “The Iwasawa conjecture for totally real fields”. In: Annals of mathematics 131.3 (1990), pp. 493–540. 118 Appendices 119 AppendixA ACoherentIntroductiontoLowerK-theory In this Appendix part, we will give a brief and coherent introduction to the foundation of the mod- ern algebraic K-theory. In particular, we will mainly examine the following three kinds of groups: K 0 (Grothendieck group),K 1 (Whitehead group), andK 2 (Steinberg group). By considering these notions, a natural extension from these K-groups to higherK n groups can then be naturally constructed, as intro- duced in the last chapter of this thesis. 1 A.1 TheGrothendieckgroupK 0 A.1.1 Groupcompletion Now in Pattern R (introduced in §2.1.2), if we start with an abelian group (i.e., G = “abelian group”), then following the same reasoning we’ll end up with f(B) = “commutative monoid” (a.k.a., “abelian monoid”, that is, an abelian group except (possibly) for inverses). More precisely, an abelian monoid is a set M endowed with an associative, commutative (binary) operation + : M × M → M as well as a distinguished identity 0 ∈ M. As an example, the setN of natural numbers is an abelian monoid under the usual+ operation with additive identity0. We can see the difference between f(B) andG is “(additive) inverse”, 2 which is also the most natural way to further restoref(B) back toG. Motivated by this process, if we start with a particular commutative 120 monoid, say, the setN as above, then the most natural way to build an inverse of every element is first doubling the group (so we may have just enough extra elements to assign), and the most natural way to double a group is by taking the direct sum of itself, namely:N× N (with the operation + defined as (m 1 ,n 1 )+(m 2 ,n 2 ) := (m 1 +m 2 ,n 1 +n 2 )∀m i ,n i ∈ N, which is implied in the definition of direct sums). Remember the purpose to doubleN was to construct an inverse− n to every n ∈ N , so we are motivated to treat the two coordinates in(m,n)∈N× N as the positive partm and negative partn, namely (m,n) corresponds tom− n∈Z(m,n∈N), whence we seeZ may be a possible candidate containing all inverse elements ofN. In particular, (0,n) should correspond to− n, i.e., the prospective inverse of n. Indeed, all such elements (0+m,n+m) should correspond to− n; more generally, (a+m,b+m) should be equal to(a,b) in the group “completion” we are building, wherem is any element inN. (That is because, correspondingly,(a+m)− (b+m) = a− b.) Therefore, we are inspired to identify(a,b) as (a+m,b+m)∀m∈N, denoted[a,b]=[a+m,b+m]. That is, we can define the groupcompletion of N asN× N/∼ , where∼ is the relation (which can be easily verified to be well-defined) identifying (a,b) as(a+m,b+m)∀m∈N. Equivalently,∼ identifies (a,b) as(c,d), denoted[a,b]=[c,d], if there exists m∈N witha+d+m=b+c+m. Hence, we have successfully constructed the completionN× N/∼ ofN (isomorphic toZ by the correspondence above), in which the identity element is[0,0] = [a,a] and the inverse− [a,b]=[b,a]. It is immediate this completion is abelian as isN. We are done. Can we generalize this process fromN to any abelian monoid and find its group completion? We can, since above we didn’t use any property unique toN. We replace all notationsN above with (or treat allN above as) any abelian monoid, and we will immediately get the generalized process and definition. Definition A.1.1 (Group completion). In general, we define the quotient group constructed as above as the group completion of M, written M − 1 M (as used in [50]) or M + (as used in [17]), with operation +:M + × M + →M + defined by [a + ,a − ]+[b + ,b − ]:=[a + +b + ,a − +b − ]∀a ± ,b ± ∈M. 121 Obviously, the notation M + makes more intuitive sense here, so we will stick to this symbol in this thesis until the meaning of M − 1 is further explored. This construction was first introduced by Alexan- der Grothendieck when he was proving the Grothendieck–Riemann–Roch theorem, so it’s also called the GrothendieckgroupK 0 (M) of the semigroupM. This construction has also set the foundation of the whole K-theory as it is today. Grothendieck used the letter “K” for “Klasse” (meaning “class” in German), and that is where all K’s in K-theory come from. By the constructions above, we have obtained a canonical monoid homomorphism: M ν −→ G(M) a7−→ [a,0] . (As introduced in §2.1, a monoid homomorphism is a function f : M → N satisfying f(m+m ′ ) = f(m)+f(m ′ ) andf(0) = 0 whereM andN are any given abelian monoids.) Moreover, the following properties are immediate by definition: PropertyA.1.2. (1)∀ abelian groupA,A + is (isomorphic to)A itself. (2) (Universal Property) Given any abelian monoid M together with the canonical monoid homomorphism ν : M → M + , any commutative abelian group A, and monoid homomorphism f: M → A,∃! (meaning “there exists a unique”) group homomorphism ˜ f: M + → A such that the following diagram commutes; i.e., f = ˜ f◦ ν : M M + ∀A ν ∀f ∃! ˜ f Proof. Given all the conditions above, let ˜ f : M + → A be any group homomorphism s.t. ˜ f(ν (a)) := f(a),∀a ∈ M (since ν (a) and f(a) are already determined in M + and A, we can always define ˜ f this way), and we always have∀a,b∈M, i.e.,∀[a,b]∈M + , 122 ˜ f([a,b] def = [a,0]+[0,b] inv. = [a,0]− [b,0]) i.e. = ˜ f([a,0]− [b,0]) ˜ f: hom ==== ˜ f([a,0] def = ν (a))− ˜ f([b,0] def = ν (b)) i.e. = ˜ f(ν (a))− ˜ f(ν (b)) def = f(a)− f(b). (Note we cannot proceed to getf(a)− f(b) f: hom ====f(a− b) sinceM is a monoid with inverses undefined, i.e.,f is not a group homomorphism.) This equation means iff :A→B is given, ˜ f withf = ˜ f◦ ν is uniquely defined. From the universal property (UP) and its proof, we can feel more clearly that the idea we define Grothendieck groups is by translating the originalM into a larger platformM + where the inverses exist and correspond to the extended inverses of the original monoidM. That is just like the idea of generalized functions (distributions) as introduced in Example 2.1.3. In addition, it’s natural to consider the converse for a known math theorem, so we have: PropertyA.1.3 ((Conversed) universal property). In contrary, any groupM + satisfying the above universal property is a group completion; i.e., suchM + can be identified with the set-theoretic quotient of M × M by the equivalence relation generated by (a,b) ∼ (a+m,b+m) as defined above. Let’s first explore a new view of Grothendieck groups using this universal property in some corollaries below, then we’ll go back to its proof. As the name suggested, the universal property is a more “universal” property, shared by many seem- ingly unrelated algebra terms, such as quotients, free groups, and exterior powers; the UP of most of these concepts have been directly used as their definitions. Here we can also use the universal property of the group completion as its definition (since by Property A.1.2 (2) and Property A.1.3 we have proved they are 123 equivalent) for the purpose of unification and simplification in mathematics. 3 Moreover, using the UP def- inition can simplify the proofs of many corresponding properties; the following two corollaries are great examples: CorollaryA.1.4 (Presentation of Grothendieck groups). M + :=F(M)/R(M)satisfiestheuniversalpropertyoftheGrothendieckgroupofmonoids,where F(M)is the free abelian group generated by the setM (with basisM) andR(M) is the subgroup ofM generated by the relations{[a+b]− [a]− [b]:a,b∈M} (in which “[]” denotes the monoid mapν as defined above). Proof. Let H := ⟨x− y− z : x = y +z ∈ M⟩. Define K 0 (M) := F(M)/H, and ν : M → K 0 (M) by ν : x 7→ x+H in the UP diagram. Now by the universal property of quotients, φ : M → G has a unique extension to ˜ φ:F(M)→A sinceH ⊆ ker ˜ φ⇐φ(x+y)=φ(x)+φ(y). Hence we may define ˜ f : K 0 (M) → A by x+H 7→ ˜ φ(x), which is unique by the uniqueness of ˜ φ. That means the UP of Grothendieck groups is now satisfied. Notice H corresponds toR(M) andK 0 (M) corresponds toM + in this corollary; we are done. Note that, by Property A.1.3,M + as defined in Corollary A.1.4 is indeed the group completion of M (that is, satiating Definition A.1.1), so this corollary also means K 0 (M)={F(M):[a+b]=[a]+[b]} as its presentation. CorollaryA.1.5 (Uniqueness). A group completionM + , if it exists, is unique (up to isomorphism). Proof. Suppose there exists another group M + satisfying the universal property of group completions, with morphism ¯ν :M →M + . Thus chooseA=:M + and we have: M M + M + ¯ν ¯ f ∃! ˜ ¯ f 124 By the universal property,∃! ˜ ¯ f :M + →M + (homomorphism), such that ¯ f = ˜ ¯ f◦ ¯ν . Likewise, we have: M M + M + ¯ f ¯ν ∃! ˜ ¯ν By the universal property,∃! ˜ ¯ν :M + →M + (homomorphism) such that ¯ν = ˜ ¯ν ◦ ¯ f. Thus ¯ f = ˜ ¯ f◦ ˜ ¯v◦ ¯ f, and so ˜ ¯ f◦ ˜ ¯ν =Id M +. Similarly ˜ ¯ν ◦ ˜ ¯ f =Id M + . Hence e ¯ν is an isomorphism. That shows a completion is unique up to isomorphism. Last, let’s go back to our proof of Property A.1.3. Proof sketch. First, by Corollary A.1.4 the Grothendieck groupM + = F(M)/R(M) where inF(M) we have ([m 1 ]+[m 2 ]+··· ) ≡ [m 1 +m 2 +··· ] modulo R(M). Therefore, any element of M + is of the form [m]− [n] for some m,n ∈ M, i.e., every element of M + is a difference of generators. Note that also means the monoid mapM × M → M + by(m,n) 7→ [m]− [n] is surjective. Then we can further obtain that [m] = [n] inM + if and only ifm+p = n+p for somep ∈ M. Hence combining the two conclusions yields what we want to prove (i.e., M + defined by the UP follows our original definition of group completions). The complete proof can be found on Page 70 in [50], but notice [50] defines group completion differ- ently from this thesis: that textbook uses the universal property as its definition. As a result, the proof in [50] can be directly borrowed here with just a little modification. In fact, most modern textbooks prefer the UP definition, but this thesis deliberately avoids that since UP definitions usually lack motivations. 4 Hence, we explore new and more coherent equivalent definitions to construct these notions as K-groups. 125 A.1.2 Grothendieckgroupofrings It is easy to extend the above definition of Grothendieck groups of abelian monoids to all commutative semigroups, since we didn’t use any special property of the identity element in our construction. But it’s not so immediate to generalize our definition of the Grothendieck groups to rings. We know by definition rings and groups (monoids) are related yet cannot be endowed simply with included relations− that is, a ring R is automatically a group under addition + but has a redundant operation× which cannot be included in this group structure; as a result, for example, a ring homomorphism has to be defined as not only f(r+r ′ ) = f(r) + f(r ′ ) ∀r,r ′ ∈ R but f(rr ′ ) = f(r)f(r ′ ) ∀r,r ′ ∈ R in order to translate analogous properties from group homomorphisms and make it a parallel structure of morphisms in the Ring category. Hence, it is not inclusive to define the Grothendieck group of a ring when only identifying the ringR as a semigroup or monoid; in other words, a ring hasn’t had its Grothendieck group in theRing category. Wouldn’t it be a good idea if we considered some monoid generated or attached to R and define the Grothendieck group ofthismonoid to be the Grothendieck group ofR? Indeed, this is a very general way of extending a definition in math, parallel to the topological extension defined at the beginning of the paper. So let’s also give it a name: translatedextension. Indeed, definitions or structures motivated by the translated extension can be found in math a.e. We briefly introduce this model here by three examples: ExampleA.1.6. As the easiest example, a numbern didn’t have a length, but the “line segment between its corresponding pointN on the number line (or in the complex plane ifn∈C in general) and the origin O” did, so we defined |n|:=|ON|, called the length or module of the numbern. Example A.1.7. A linear transformation T : V → W between vector spaces over a field didn’t have a rank, but its corresponding matrixA did, sorank(T):=rank(A) is then naturally defined. 5 126 ExampleA.1.8. As shown in Example 2.1.3, a distributionT ∈D ′ (Ω) is not a number so it can never be 0 in particular. But T(φ) is a number and can equal 0. So by the translated extension model, we define T :=0 on an open setU ⊆ Ω ifT(φ)=0∀φ∈D(U). By this translated extension model, the first natural way to obtain an abelian monoid (group) from a ringR may be theR-module (which is an abelian group equipped with scalar multiplication: R× M → M by(r,m) 7→ rm). Hence, here we can choose “the monoid generated or attached to R” to be the monoid consisting of isomorphism classes of finitely-generated projective modules over the ring R, that is,Iso(R-Proj). Hence, we have deduced the following definition: DefinitionA.1.9 (Grothendieck group of a ring). IfR is a ring, theGrothendieckgroupK 0 (R) is defined by K 0 (R):=K 0 (Iso(R-Proj)) whereR-Proj is the full subcategory generated by all finitely generated projective left R-modules. Note thatR-Proj is avirtuallysmall category, meaning its familyIso(R-Proj) of isomorphism classes has a cardinal number. But why do we specially designK 0 (R) by thisIso(R-Proj) monoid instead of other monoids related toR? The reason is, as analyzed above, it also defines a functor on the category Ring; indeed we have: TheoremA.1.10 ([41]). There exists a functorK 0 : Ring → Ab withR 7→ K 0 (R); ifφ : R → S is a ring homomorphism, then K 0 (φ):[R n ]7→[S n ] ∀n⩾0, meaningK 0 (φ) takes freeR-modules to freeS-modules. For example, now we have defined K 0 (Z) in two different ways: K 0 (Z)            Def A.1.1 ∼ = Z by viewingZ as an abelian monoid; Def A.1.9 = K 0 (Iso(Z-Proj)) by viewingZ as a ring. 127 HereZ-Proj is the semigroup of isomorphism classes of finitely-generated projective Z-modules (i.e., free abelian groups). A nice extended definition should agree with the original one in the domain where the subjects intersect, namely in this case we should haveK 0 (Iso(Z-Proj)) ∼ =Z. So we explore and obtain the following proposition: TheoremA.1.11 ([41]). If R is a ring having a rank function (that is, R m ∼ = R n implies m = n), and all of its finitely generated projectiveR-modules are free, thenK 0 (R) (as in Definition A.1.9) ∼ =Z. It is immediate to deduce: CorollaryA.1.12. K 0 (R) ∼ =Z ifR is a division ring, a PID, a local ring, orF [x 1 ,...,x n ] when F is a field. Now we know, by this corollary, we do haveK 0 (Z) Def A.1.9 ==== K 0 (Iso(Z− Proj)) ∼ =Z sinceZ is a PID. Warning: In this case, the group completion map γ : Iso(R− Proj) → K 0 (R) is frequently not injective, meaning there might usually be some rings having identical Grothendieck groups, i.e., group completions (see [17]). A.2 TheWhiteheadgroupK 1 In 1962, Hyman Bass and Stephen Schanuel [5] introduced the first complete definition of the first K-group of a ring: DefinitionA.2.1 (First K-group of a ring). For any ring R, thefirstK-group K 1 (R) is defined by K 1 (R)=GL(R) ab :=GL(R)/GL ′ (R) 128 In the equation, GL(R) := lim −→ GL n (R) is called the infinite general linear group (the set of infinite- order invertible matrices), where lim −→ GL n (R):=⊔ i GL n (R)/∼ is the direct limit (colimit) of GL n (R), which embeds inGL n+1 (R) as the upper left block matrix. AndGL ′ (R):=[GL(R),GL(R)] denotes the commutator subgroup ofGL(R). As we can see, this so-called K 1 is somewhat related to K 0 (which denotes the completion of some monoid); that is, either of them can be viewed as the quotient group of some new group (the free group F(K 0 (Iso(R-Proj))) forK 0 andGL(R) forK 1 ) generated by the ringR. We’ll explore further connec- tions in the last chapter of the main body of the thesis. By definition, obviously K 1 (R) is always an abelian group; indeed, it is specially designed to be the abelianization of the infinite general linear group. It is also the maximal abelian quotient group ofGL(R), since in general:∀N⊴G,G/N is abelian⇔G ′ ⊆ N. In linear algebra, we have known thatGL n (R) is generated by elementary matrices (including row- switching transformations, row-multiplying transformations, and row-addition transformations) if R is any Euclidean domain (Page 677 in [41]). Now let’s first consider row-addition transformations e ij (r), 6 which refer to the matrices with at most one non-zero diagonal element and with all diagonal elements equal to 1− more specifically, matrices of the following form, where r is in the(i,j)-spot:              1 0 . . . 1 r . . . 1 . . . 0 1              In other words, they are the matrices which may be reduced to the identity matrix I by only row operations, denotedR ij (r), of addingr times rowj to rowi. Thus, “{the subgroup ofGL n (R) generated by elementary matrices e ij (r) with 1 ⩽ i,j ⩽ n} =: E n (R)” is not proper in general. Moreover, if R has stable ranged+1 thenE n (R) is always a normal subgroup ofGL n (R)∀n⩾ d+2; still further, if 129 R is commutative then E n (R)⊴GL n (R)∀n⩾ 3 (Page 200 in [50]). Thus, it is natural to consider the quotientGL n (R)/E n (R), where we equate all matrices that can be reduced toI by onlyR ij (r). Imitating the relation of GL(R) and GL n (R), we may also define E(R) := {the (normal) subgroup of GL(R) generated by the elementary matricese ij (r)} and examineGL(R)/E(R). We have: TheoremA.2.2 (Whitehead’s lemma). The commutator subgroupGL ′ (R)(⊆ GL(R)) equalsE(R). Equivalently,K 1 (R)=GL(R)/E(R). Proof sketch. Note here “equal” does not just mean “isomorphism” but real “equal”, which enables us to use the common trick: to show two sets are equal by proving two inclusion relations both hold. First, for n ⩾ 3, by the fact that e ij (r) = [e ik (r),e kj (1)] ∀i,j,k distinct, we see that E n (R) = [E n (R),E n (R)]; that is, E n (R) is a perfect group∀n ⩾ 3. Thus the commutator subgroup GL ′ (R) ⊇ E(R). Conversely, every commutator[g,h]∈GL ′ n (R) can be expressed as a product inGL 2n (R): [g,h]=     g 0 0 g − 1         h 0 0 h − 1         (hg) − 1 0 0 hg     , and we easily see each of these terms∈E 2n (R). ThusGL ′ (R)⊆ E(R) by direct limit. ThereforeGL ′ (R)=E(R). In fact, the group GL(R)/E(R), called the Whitehead group of the ring R, was first defined by Whitehead in his paper [52]. Hence, by this lemma, we may also call K 1 (R) the Whitehead group of R. Now that we have defined K 1 of a ring, by the translated extension model, many definitions in this category come out naturally. 7 As an example, A group G didn’t have a whitehead group, but its group ringZ[G] does (i.e.,K 1 (Z[G])), so we may define the “Whitehead group of G” asK 1 (Z[G])/G, that is, the cokernel ofG×{± 1}→K 1 (Z[G]). 130 We know closely related toGL n (F) is the special linear groupSL n (F):={A∈GL n (F):det(A)= 1}⊴ GL n (F), for any field F ; equivalently, SL n (F) := kerdet, where det : GL n (F) → F × := {all invertible elements in F } (i.e., the multiplicative group of nonzero elements, with its identity 1) is the determinant surjective (since∀a ∈ F × , det :     a 0 0 I     n× n 7→ a where I is the identity matrix) homomorphism. In particular, ifF =R, then by Prop 7.16 in [27], we further haveSL n (R) is a properly embedded Lie subgroup of dimensionn 2 − 1. Now let’s extend the definition above from the pair “ (F,GL n )” to “(any commutative ring R, the infinite general linear group GL)”. Likewise, we havedet : GL(R)→ R × := {all invertible elements in R} by the same map, andSL(R) := kerdet as before. By definition, we have detE(R) = 1 (soE(R)⊆ kerdet), and furtherE(R)⊴SL(R) i.e. = kerdet (sincek∈E(R) andg∈SL(R) implygkg − 1 ∈E(R)). So the map det descends to (meaning “inducing”) K 1 (R) ThA.2.2 ==== GL(R)/E(R) → R × , which we denote as det/E. Hence, compared to the previous definition of SL(R) := kerdet, a new definition comes out naturally: SK 1 (R) := ker(det/E). We also have SK 1 (R) def = ker(det/E) = (kerdet)/E(R) def = SL(R)/E(R) by definition of kernels. 8 In summary, we have obtained: ker(det/E) def = SK 1 (R)=SL(R)/E(R) ϕ →GL(R)/E(R) ThA.2.2 ==== K 1 (R) def = GL(R)/GL ′ (R) det/E → R × . In the above sequence,ϕ is the canonical injection fromSL(R)/E(R) toGL(R)/E(R). And we can see the map det/E is surjective since it obviously splits (“R × can go back to K 1 (R)”) via the general linear group of degree 1 (which can be viewed as numbers) through the natural mapR × → GL 1 (R)→ K 1 (R). Hence, we have constructed the following split short exact sequence (since we’re talking about multiplicative groups, we use 1 instead of 0 to denote the identity): 1→SK 1 (R) ϕ →K 1 (R) det/E → R × →1. (A.1) 131 Therefore we have obtainedK 1 (R)/imϕ ∼ =R × , andK 1 (R) ∼ =SK 1 (R)⊕ R × . A.3 TheSteinberggroupK 2 Let’s take another look at these elementary matrices (namelye ij (r)) defined above. We can easily find and verify some basic properties by definition of matrix inverse and multiplication: e ij (r)e ij (s)=e ij (r+s); (A.2) [e ij (r),e kℓ (s)]=                1 ifj̸=k,i̸=ℓ e iℓ (rs) ifj =k,i̸=ℓ e kj (− rs) ifj̸=k,i=ℓ . (A.3) Here[x,y] := xyx − 1 y − 1 (in general) is the commutator ofx andy, and the matrix ordern⩾ 3. Also, by definition, E n (R) is generated by these elementary matricese ij (r) (with1⩽i,j⩽n). We find these relation equations (A.2) and (A.3) are very much like something unique to elementary properties which may be considered “core”, so let’s again do the topological extension and see what we will get: DefinitionA.3.1 (Topological extension of elementary matrices). 9 Given a ringR and any setA of ele- ments x ij (r) where r ∈ R and 1 < i,j ⩽ n for n⩾ 3 (notice x ij (r) are not necessarily elements in a matrix), ifx ij (r) satisfy x ij (r)x ij (s)=x ij (r+s) (A.2 ′ ) [x ij (r),x kℓ (s)]=                1 ifj̸=k andi̸=ℓ x iℓ (rs) ifj =k andi̸=ℓ x kj (− sr) ifj̸=k andi=ℓ (A.3 ′ ) 132 for allr,s∈R, then we callx ij (r)elementarymatrices (in the “topological extension” sense). Further- more, we call the group generated by these elementary matricesE n (R). Obviously, whenA denotes a matrix inGL n (R),E n (R) defined above agrees with our original defi- nition ofE n (R). Moreover, with the name “elementary matrices” and “E n (R)” in the extension sense, we are enabled to consider every corresponding notion, such as SL(R),GL(R),K 1 (R), etc., related to the original “elementary matrices”, based on the topological extension model discussed in the main body of the thesis. One can even further deduce more definitions following PatternR. In fact, the group generated by these elementary matrices defined above is usually called the (unstable) SteinberggroupSt n (R) ofR, and the relations (A.2 ′ ) and (A.3 ′ ) are called theSteinbergrelations. 10 In other words, the Steinberg group is the free group on generators divided by the Steinberg relations. That is,St n (R) :=⟨x ij (r),1 < k,j⩽ n|R⟩, wherer∈ R, and the boldR refers to the Steinberg relations (A.2 ′ ) and (A.3 ′ ) given in Definition A.3.1. As in the previous section, it is natural to define St(R) as the direct limit ofSt n (R), called the(stable)Steinberggroup ofR. Another motivation to build Definition A.3.1 was to explore the first K-group in a new way. Based on our previous definitions, let’s consider the canonical map φ:St(R)→GL(R) defined by x ij (r)7→e ij (r), which is ontoE(R), i.e.,φ:St(R)↠ E(R)⊆ GL(R). It is easy to find: cokerφ def = GL(R)/imφ(where imφ def ofφ ====E(R))=GL(R)/E(R)=:K 1 (R). That provides us with another possible approach to define K 1 (R). Now we’ve known the cokernel, so what about the kernel ofφ? The kernel is obviously not trivial and is not identical for eachR; hencekerφ has the potential to be formed as a new definition: Definition A.3.2 (Second K-theory group). We set K 2 (R) := ker(φ : St(R) → GL(R)) for a ring R, called thesecondK-theorygroup. 133 Why do we call itK 2 ? Not only because it is related withK 1 in a kernel-and-cokernel way, but there are some other (or “further”) reasons which are explored in the main body of the thesis. Let’s point out thatK 2 (R) = ker(φ : St(R)↠ E(R)) also, by definition. Indeed, we have deduced the following exact sequence of groups: 11 1→K 2 (R),→St(R) φ ↠ E(R)⊆ GL(R)↠ K 1 (R)→1. (A.4) Moreover, one can find: TheoremA.3.3 ([40]). K 2 (R)=Z(St(R)), i.e.,K 2 (R) is precisely the center ofSt(R). First, in particular, we obtain K 2 (R) is always an abelian group, as was K 1 (R). Also, by definition, 1→ K 2 (R)→ St(R)→ E(R)→ 1 is a central extension ofE(R) byK 2 (R); shortly, we saySt(R) or St(R) → E(R) (not the whole exact sequence) is a central extension. Here “central”, of course, means the subgroup K 2 (R) lies in the center of St(R). As for the phrase “extension of E(R) by K 2 (R)”, in fact, the idea of “central extension” is partially motivated and regularly used in considering homotopy classes in homotopical algebra, and it’s natural to say a central extension is of the “base” by the “fiber”. As a result, we sometimes extend this idea to all extensions in general; that is, in the short exact sequence 0→A f →B g →C→0, we may call this sequence an extension of C by A. However, the same sequence can also be called an extension of A by C in the literature, for example, in [41]. This latter naming is motivated and based on different origins, such as Galois theory. 12 Indeed, this extension is further auniversal central extension ofE(R), meaning this central extension is “unique” (up to isomorphism): 134 Definition A.3.4 (Universal central extension). In general, a universal central extension of G is a central extensionX π −→ G such that∀ other central extensionY τ −→ G,∃! homomorphismf : X → Y such thatπ =τ ◦ f. PropertyA.3.5 ([50] 13 ). (1) Every central extensionY π −→ St n (R) is split for alln⩾5. (2) (Recognition theorem) Every perfect groupG (i.e.,G=[G,G]) has a universal central extension: 1→H 2 (G;Z)→[F,F]/[R,F]→G→1 Moreover, the following are equivalent providing thatX is a central extension ofG : (i)X is universal; (ii)X is perfect and every central extension ofX splits; (iii) The first and second homology groups of X both vanish, that is,H 1 (X;Z)=H 2 (X;Z)=0. (3) By (1) and (2), it is immediate thatSt n (R) is the universal central extension ofE n (R). (4) (Kervaire, Steinberg) By (3), we getSt(R) is a universal central extension ofE(R). Hence, for the same “universal” reason as described in §A.1.1 (Group completion), sometimes we di- rectly define the Steinberg group (of a ring R) as the universal central extension of E(R) (which is iso- morphic to the commutator subgroup ofGL(R) by Whitehead’s lemma). This long property has a famous corollary: K 2 (R) ∼ = H 2 (E(R);Z), whereH 2 (E(R);Z) is the Schur multiplier of groupE(R), namely the homology group ofE(R) (a.k.a., the second homology module of E(R) with coefficients in Z; see Definition 6.4.1). The proof is immediate from (2) and (4) in Property A.3.5. Moreover, we noticeE(R)=[GL(R),GL(R)] by Whitehead’s lemma, soK 2 (R) ∼ =H 2 ([GL(R),GL(R)];Z). 135 AppendixB Notes Chapter1 1. Naturally we would ask: isP(1) the “simplest" subseries of the harmonic series to be divergent? That is, is every subsequence of P(1) convergent? If not, until what subsequence of P(1) would we find the simplest form? Does it exist? Can we find a generalized theorem, such as the “rank" of series, to categorize when a general series can achieve its simplest form? We see all primes generate natural numbers, so does this fact have to do with the rank? Chapter2 1. Unfortunately, to my knowledge, except for some historians, very few mathematicians really care about the philosophy or motivation behind a kind of math naming like even/odd functions, much less compare this naming with “open sets” and generalize them to get a universal model like the topological extension. I believe such philosophies behind math definitions and conventions will play an increasingly essential role as mathematics proceeds. 2. The topological extension and the translated extension (as introduced in Appendix A) are original ideas in this thesis. 3. It’s also written asC ′ :=C\{1}. 136 Chapter4 1. Namely in terms of the subset symbols, we have{modern math}={clear math}/ ∼ = where ∼ = identifies (⊂ ,⊆ ) with(⊊ ,⊂ ). 2. Whatever the reasons are, I don’t think abuses like this will disappear over time until the year when new math subjects force mathematicians to strictly care about the differentiation (like “ =” for sets). Chapter6 1. What motivated us to explore this metric-independent property is, of course, the invention of topological space in general as well as the topological extension model. Once we substitute this extension toPattern R, then I believe a new (NOT up to “isomorphism”)* math subject parallel (at least isomorphic) to point-set topology will be made possible. *Note: “new up to isomorphism” means “truly new”, otherwise it might become not new once we identify all isomorphisms. 2. It’s equivalent to defining the empty set if we define something that doesn’t exist, which is usually not significantly constructive in math. 3. Sometimes “algebraic number field” is also called “ arithmetic number field" or simply “number field”. 4. As we usually use the universal property as the abstract definition (see Appendix A), we also usually use each item of this Property 6.3.3 as the equivalent definition of Dedekind domains. 5. We know every nonzero prime ideal of a PID is maximal. 6. As we can see, this generalization is not considered a topological extension since it doesn’t directly use any properties as a definition in a bigger universe. 7. If I had found such a connection, I would have used this more natural way to define K i in my current thesis and avoided letting the more complicatedBGL(R) + appear by starting with the homotopy groups. (That said, maybe this new homology pattern will also consist ofπ i somewhere, if the pattern exists.) 137 8. Interestingly, when I googled “Differential K-theory” I found this term was already defined, but it refers to a marginal psychology and criminology subject having nothing to do with math or our K-theory. Their “K” comes from ther/Kselectiontheory instead of Grothendieck. ChapterA 1. In the appendix, I tried to especially focus on motivations behind all the thoughts and ideas. To be exact, we did not follow the conventional methodology that used to be adopted by the Bourbaki School—that is, (taking Example 2.1.3 as an example) “here’s the definition of distributional derivatives, so next let’s form a theorem proving(D α T f )(φ) = T D α f (φ); by this theorem, it follows that the distributional derivative coincides with the classical derivative.” At least this is not how mathematicians invented the weak deriva- tive. In my opinion, this kind of description is just like a sole tree crown hanging in the sky, without the trunk or root; hence this tree has little potential to keep growing. Rather, abstractness is always supported by concreteness. In most cases, there are certain impetus and motivations (like what our topological ex- tension model concluded) serving as the “root” that supports everything. With them, a complete tree with vitality and the ability to continue growing will be obtained, which provides a future for mathematics. This is the central idea of the word “coherent” in the name of the appendix, and this idea is what I tried to concentrate on in writing this part of the thesis. 2. We may as well write it asR− G={inverses} orG+{inverses}=R. 3. However, being more abstract usually means less intuitive and so has less potential to be further devel- oped in a meaningful way, unless we go back to the basics and understand all the motivations. Therefore, for most concepts, this appendix does not adhere to the usual definitions found in most papers or textbooks but adopts a more coherent, intuitive approach with all the motivations clearly explained. 4. Although UP definitions are undoubtedly more universal and concise, they are much less intuitive and less friendly to many readers. 138 5. Sometimes we first define rank(T) := dim(ImT) followed by defining rank(A) in order to make the definition more abstract (unified) in abstract algebra, but in this example I followed what I believe is the most intuitive way: I assume we defined rank(A) first as the number of non-zero rows of the corresponding simple matrix ofA, namely, the dimension of its row (column) space. 6. It is noteworthy that usually we directly calle ij (r) “elementary matrices” for convenience (this is also a topological extension), which is actually only a proper subset of what we used to call elementary matrices in standard linear algebra. 7. Through our inference in this part, again, we can see that any abstract definition follows some strict reasoning and certain motivations; they should come out very naturally instead of being the “free group” generated by arbitrary generators—we need relations to restrict them from growing wild, and these rela- tions are our motivations. This is also a life philosophy that I embrace: we need self-discipline and hard work to obtain the “locally optimal solution” of life. 8. As almost no math textbooks or papers do, I differentiate these two symbols in my paper to avoid misunderstandings: def = (or △ =) meaning “equal to by (because of) definition”, and := or=: (where the: side is the new definition) meaning “defined to be equal to”. By this interpretation they are two very different symbols where the first means both sides of the symbol are already equal based on the prerequisite logic while the second means we proactively let both sides equal in this step by defining them to be equal. I think it’s good to simplify notations in math papers, but in textbooks such identifications really confuse learners since most textbook users are not math experts in this field. 9. This definition is originally constructed in this thesis, namely without formal references. As we can see, this definition is designed to be parallel with the naming of open sets. 10. In reality, Steinberg first extracted these relations from elementary matrices, but he didn’t proceed to define more related notions as in PatternR. 139 11. Of course, the sequence “1→K 2 (R)→St(R)→··· ” being exact has implied “K 2 (R),→St(R)→ ··· ”, and the same for the surjection part on the right. I put the hooked and two-headed arrows here only for emphasizing. 12. Intuitively, in this example of the second K-group, the mapSt(R)→E(R) is surjective, namelySt(R) is “bigger” thanE(R), so we can callSt(R) an extension ofE(R). On the other hand,K 2 (R)→ St(R) is injective, so it also makes sense when we callSt(R) an extension ofK 2 (R). 13. Theorems like this (followed by references) are reorganized (rather than directly quoted) from their references, but the basic ideas are the same so we omit their proofs in this thesis. 140

Abstract (if available)

Abstract In this thesis, the Riemann zeta function is introduced first through the sieve of Eratosthenes and product formulas, by which its relationship with prime numbers is illustrated. Following this, the analytic continuation of the Riemann zeta function, as well as the Hurwitz zeta function, is discussed from two different perspectives: contour integration and functional equations. Based on these foundations and the construction of similar zeta functions, numerous results, such as the Basel problem, Bernoulli polynomials, and Apéry’s constant, are presented about special values of the Riemann zeta function, including the distribution of its zeros. Lastly, some concepts of algebraic K-theory and étale cohomology are outlined, whereby a connection between the Riemann zeta function and K-groups of rings of integers is demonstrated. Moreover, the Riemann zeta function is further generalized to the local zeta function and the Dedekind zeta function in the classification of K-groups of the ring of integers. A brief introduction to lower K-groups can be found in Appendices, which serves as a prerequisite to understanding the chapter about K-theory. Also included in this thesis is an original concept called the topological extension model that connects many seemingly unrelated concepts about zeta functions and K-theory.

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Creator Feng, Zhanhu (author)

Core Title Zeta functions: from prime numbers to K-theory

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Mathematics

Degree Conferral Date 2022-05

Publication Date 04/16/2022

Defense Date 03/11/2022

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Tag algebraic K-theory,analytic continuation,analytic number theory,Bernoulli polynomials,OAI-PMH Harvest,Riemann zeta function

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Advisor Friedlander, Eric (committee chair), Kamienny, Sheldon (committee member), Montgomery, Susan (committee member)

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