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On the Hilbert-Pólya and Pair Correlation Conjectures Hudoba de Badyn, Mathias 2014-01-01

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The University of British ColumbiaOn the Hilbert-Po´lya and Pair Correlation ConjecturesAuthor:Mathias Hudoba de BadynSupervisor:Dr. Greg MartinA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFBACHELOR OF SCIENCECOMBINED HONOURS IN PHYSICS AND MATHEMATICSAuthor DateSupervisor Date2c©2013 Mathias Hudoba de Badyn1The complexities of the Riemann Zeta Function. [19]Contents1 Introduction 51.1 Motivation for Studying ζ(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The Euler Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 ζ(s) and ξ(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 The Conjecture of Hilbert and Po´lya, and of Montgomery . . . . . . . . . . . 112 Riemann’s Wonderful Function 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Prime Counting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Perron’s Formula and Mellin Transform Theory . . . . . . . . . . . . 162.2.2 First Proof of ψ(x) Formula . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Second Proof of ψ(x) Formula . . . . . . . . . . . . . . . . . . . . . . 202.3 Density of the Non-Trivial Roots . . . . . . . . . . . . . . . . . . . . . . . . 223 A Primer on Physics 233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Operators in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 293.3.3 Matrix Representation of an Operator . . . . . . . . . . . . . . . . . 302CONTENTS 33.3.4 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . 314 The Hilbert-Po´lya Conjecture 334.1 Introductory Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1 Gaussian Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Pair Correlation Function for Gaussian Unitary Matrices . . . . . . . 344.3 Zeroes of ζ(s) and Random Matrices . . . . . . . . . . . . . . . . . . . . . . 364.4 The Berry-Keating Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Bibliography 43Appendices 45A Appendix 46AbstractThe Hilbert-Po´lya Conjecture supposes that there exists an operator in a Hilbert space whoseeigenvalues are the zeroes of the Riemann Zeta function ζ(s). This conjecture, if true, would verylikely expedite the proof of the Riemann Hypothesis, namely that the non-trivial zeroes of ζ(s)have real part 12 . In this thesis we summarize work by Berry, Keating and others in constructingsuch an operator. Although the work so far has not yet yielded such an operator, some have beenfound that have properties very close to what is desired. We also summarize a (partially proven)conjecture by Montgomery that motivates the search for this operator. He conjectures that thepair correlation function for the spacing between the imaginary parts of the Riemann zeroes is thesame as the correlation function for the spacing between eigenvalues of random Gaussian unitarymatrices.AcknowledgementsI would like to first thank my parents for the many things they have done that have helped contributeto my successes so far, one of which was their decision to give me a name which contains ‘math’as a subsequence. I would also like to thank the laws of physics and mathematics which allowthe Universe to exist the way it does and for allowing the atoms which comprise my body to haveCONTENTS 4evolved over the last 13.something billion years to the point where they can begin to comprehendsaid laws in the form of the biological computation unit that is me.I must also give thanks to my supervisor Dr. Greg Martin for his patience, guidance anduniformly continuous support, and also his excellent clarity and wit which made the study of thematerial in this thesis all the more interesting. There have also been several other mathematiciansand physicists at the University of British Columbia which have helped me a a great deal in myundergraduate years. I must thank Dr. Joanna Karczmarek for supervising my concurrently-written physics Honours thesis in string theory, as well as Dr. Fok-Shuen Leung for mentoring mein my first year in the Science One program, and Dr. Brian Marcus for advising my junior seminarcourse in entropy, information theory and graph theory. I also thank all the professors in both thephysics and mathematics departments who have taught the various courses I have taken (and haveyet to take). In particular, I must thank Dr. David Morrissey, Dr. Mark Van Raamsdonk and Dr.Ariel Zhitnitsky for teaching me elementary quantum mechanics, and quantum field theory whichis relevant in Chapters 3 and 4 of this thesis.I should also acknowledge the ‘support’ of my peers, in particular my fellow students in theCombined Honours in Physics and Mathematics program, past and present, and the HonoursMathematics program. In alphabetical order (I think), these are Daniel Baker, Kahan Dare, DeshinFinlay, Andrew Kuba Karpierz, Paul Liu, Kevin Martin, Emily Neufeld and Kai Ogasawara. Inparticular, I should thank Justin Scarfy for lending me his copy of Edwards. I also thank the manyother students in Honours Physics, and Combined Honours in Physics with [not math], who aretoo numerable to list here.Chapter1IntroductionWith great power, comes greatprivilege. I mean responsibility.Not Voltaire1.1. Motivation for Studying ζ(s)In Bernhard Riemann’s classic paper On the Number of Primes Less Than a Given Mag-nitude [15], he attempted to find a formula for pi(x), the function whose value at x is thenumber of prime numbers in the interval [1, x]. In the paper, he defined the seeminglyinnocent function:Definition 1 (The Riemann Zeta Function). Let n ∈ Z and s > 1. The Riemann ZetaFunction is defined as:ζ(s) =∞∑n=11ns. (1.1)This was in part motivated due to a conjecture by Gauss, namely that this prime countingfunction behaves aspi(x) ∼xlog x. (1.2)This was later formally proven by Hadamard and de la Valle´e Poussin as the Prime NumberTheorem:5CHAPTER 1. INTRODUCTION 6Theorem 2 (The Prime Number Theorem).Let the prime counting function be given by pi(x). Then,limn→∞pi(x) log xx= 1. (1.3)Equivalently, as x goes to infinity,pi(x) ∼xlog xThe proof may be found in Chapter 4 of [8]. The connection between the Riemann Zetafunction and prime numbers lies primarily in the Euler Product.1.2. The Euler Product FormulaTheorem 3 (The Euler Product Theorem). Let the Riemann Zeta Function beζ(s) =∞∑n=11ns.Then,ζ(s) =∏p11− p−s, (1.4)where p denotes prime numbers, and <(s) > 1.Proof. Note thatζ(s) = 1 +12s+13s+14s+ . . .and12sζ(s) =12s+14s+16s+18s+ . . . .The right-hand side of the second line contains all of the terms of the Riemann Zeta functioncorresponding to the integers that have 2 as a factor. Subtracting these two equations yields(1−12s)ζ(s) = 1 +13s+15s+17s+19s+ . . . .The right-hand side of this equation now has all the terms from ζ(s) with factor 12s removed.Note that these right-hand sides are all absolutely convergent Dirichlet series for <(s) > 1,CHAPTER 1. INTRODUCTION 7since the numerators are all 1, and thus bounded. Now, repeat the multiplication andsubtraction steps for 13s :13s(1−12s)ζ(s) =13s+19s+115s+ . . . .(1−13s)(1−12s)ζ(s) = 1 +15s+17s+111s+113s+ . . .We can see that all numbers in this infinite series that have a factor of 2 or 3 are ‘sieved’in a manner like the Sieve of Erastosthenes. By the Fundamental Theorem of Arithmeticevery number that is not prime can be decomposed into prime factors, and hence repeatingthis algorithm for all primes p ad infinitum yields:ζ(s)(1−12s)(1−13s)(1−15s)· · · = 1.The convergence of the right-hand side follows from the absolute convergence of the Dirichletseries for <(s) > 1. It is now clear thatζ(s) =∏p11− p−s, (1.5)however we must now show that the right-hand side of 1.5 converges. Considerlog ζ(s) = log(∏p11− p−s)= log(11− 2−s)+ log(11− 3−s)+ log(11− 5−s)+ . . .= − [log(1− 2−s) + log(1− 3−s) + log(1− 5−s) + . . . ]CHAPTER 1. INTRODUCTION 8We claim that this is absolutely convergent.| log(1− 2−s) + log(1− 3−s) + log(1− 5−s) + . . . |≤ | log(1− 2−s)|+ | log(1− 3−s)|+ | log(1− 5−s)|+ . . .≤∣∣∣∣122∣∣∣∣+∣∣∣∣13s∣∣∣∣+∣∣∣∣15s∣∣∣∣+ . . .≤∣∣∣∣11s∣∣∣∣+∣∣∣∣12s∣∣∣∣+∣∣∣∣13s∣∣∣∣+ . . . ,where the bound | log(1 + x)| ≤ |x| is given from Equation 2 in [18]. Hence, by the absoluteconvergence of a Dirichlet series with <(s) > 1 yields the convergence of the Euler productfor <(s) > 1.This theorem should make clear that there is indeed a very intimate relationship betweenprime numbers and ζ(s). In chapter 2, we utilize this relationship to outline the constructionof pi(x) from ζ(s). For now, we discuss some properties of the Riemann Zeta function,including its convergence, analytic continuation and its relationship with its ‘sister’ functionξ(s). In the last section of this chapter, we give a brief overview of the Hilbert Po´lya andPair Correlation Conjectures, which are the main topics of this thesis.1.3. ζ(s) and ξ(s)We can write the Riemann Zeta function in implicit functional form, and relate it to afunction ξ(s). This will be useful in further discussions in Chapter 2. First, recall thedefinition of the Gamma Function.Definition 4 (The Gamma Function). Let s be a complex number with real part greater than0. DefineΓ(s) =∫ ∞0xs−1e−xdx. (1.6)Then, via integration by parts, we haveΓ(s+ 1) = sΓ(s). (1.7)CHAPTER 1. INTRODUCTION 9Noting that Γ(1) = 1, we can see that for non-negative integer values of Γ, we have thatΓ(n) = (n− 1)!. Via coordinate substitution, we can transform x→ pin2x for integer n intoEquation 1.6 to getpi−s2 Γ(s2)n−s =∫ ∞0xs2−1e−n2pixdx. (1.8)Next, we can sum both sides over all positive integers n.∞∑n=1pi−s2 Γ(s2)n−s =∞∑n=1∫ ∞0xs2−1e−n2pixdx. (1.9)giving a factor of ζ(s) on the left-hand-side. Note that by doing this, we lose convergencefor <(s) ∈ (0, 1). Thus, <(s) > 1. On the right-hand side, we have absolute convergenceof the sum, so we can interchange the order of summation and integration. We now state alemma that will lead us towards defining a function ξ(s) which we will show to have someinteresting and useful properties related to ζ(s).Lemma 5 (A Lemma Pertaining to ζ(s)). Define the useful functionφ(x) =∞∑n=1e−n2pix. (1.10)and let Γ(s) be defined as before. Then,pi−s2 Γ(s2)ζ(s) = −1s(1− s)+∫ ∞1(xs2−1 + x1−s2 −1)φ(x)dx (1.11)Proof. Following the proof in [16], the φ(x) functionφ(x) =∞∑n=1e−n2pix (1.12)will givepi−s2 Γ(s2)ζ(s) =∫ ∞0xs2−1φ(x)dx. (1.13)We can now split the integral on the right-hand-side into an integral from x ∈ (0, 1] andx ∈ [1,∞):pi−s2 Γ(s2)ζ(s) =∫ 10xs2−1φ(x)dx+∫ ∞1xs2−1φ(x)dx. (1.14)CHAPTER 1. INTRODUCTION 10Substituting x → 1x in the integral from x ∈ (0, 1] yields an integral from x ∈ [1,∞). Thisis justified because the integral is still convergent under this coordinate transformation. Weget:pi−s2 Γ(s2)ζ(s) =∫ ∞1x−s2−1φ(1/x)dx+∫ ∞1xs2−1φ(x)dx. (1.15)With some work, one may derive from Equation 1.10 thatφ(1x)=√x2−12+√xφ(x). (1.16)Substituting this into the integral in Equation 1.15 and integrating out the constant termswill give:pi−s2 Γ(s2)ζ(s) = −1s(1− s)+∫ ∞1(xs2−1 + x1−s2 −1)φ(x)dx (1.17)The right-hand side of this equation is in fact meromorphic on C, since the integralconverges absolutely for all complex numbers due to the behaviour of φ(x) as x approachesinfinity. We can bound φ(x) asφ(x) =∞∑n=1e−pin2x <∞∑n=1e−(pix)n =1epix − 1(1.18)Therefore, φ(x) behaves as O(e−pix) as x approaches ∞ and imposes absolute convergenceon the integral in Equation 1.17. Furthermore, the right-hand-side is invariant under thetransformation s→ 1− s. From this, we can define the functionξ(s) =12s(s− 1)pi−12 Γ(12)ζ(2), (1.19)which from Equation 1.17 satisfies:ξ(s) = ξ(1− s). (1.20)The ξ(s) function is entire for <(s) > 0, since the factor of (s− 1) cancels the pole of ζ(s) ats = 1. We can also see from this definition that there are zeros of ζ(s) at the poles of Γ(s/2)at the negative even integers (the factor of s counters the pole at s = 0). These are referredCHAPTER 1. INTRODUCTION 11to as the ’trivial zeros’, in that they are trivial due to rather well-understood behaviour ofΓ(s). A colour plot of the Riemann zeta function (colours corresponding to the argument ofζ(s)) can be seen in Figure 1.1.We are now in a position to state the famous hypothesis of Riemann about the other(non-trivial) zeros of ζ(s).Conjecture 6 (The Riemann Hypothesis). Let ζ(s) be defined by some analytic continuationon C (perhaps Equation 1.17. Then, all non-trivial zeros of ζ(s), namely those that do notcome from the poles of Γ(s/2), have real part 12 .Hopefully one day we (I) will also be in a position to prove Conjecture 6.1.4. The Conjecture of Hilbert and Po´lya, and of MontgomeryThe Hilbert-Po´lya conjecture is one method of tackling the problem of proving or disprovingthe Riemann Hypothesis. The conjecture is as follows.Conjecture 7 (The Hilbert-Po´lya Conjecture). There exists a self-adjoint operator in aninfinite-dimensional Hilbert space whose eigenvalues are the non-trivial Riemann zeros.If this conjecture is true, then it would be a (presumably) relatively simple transitionto equate the Riemann Hypothesis to a statement about the eigenvalues of this operator(perhaps other than the obvious statement ‘the real part of the eigenvalues is 12 ’). Sincelinear algebra has been extremely well studied in the past century, in part due to the adventof quantum mechanics and the need for engineers to study and solve large systems of linearequations, it is a promising idea that reducing the Riemann Hypothesis to a statement aboutthe eigenvalues of this operator would simplify the proof or disproof significantly. Althoughnot formally discussed in this thesis, Connes has created a trace formula equivalent to theRiemann Hypothesis.Montgomery was studying the pair correlation between spacings of the zeros of the Rie-mann Zeta function. Upon presenting his work at the Institute for Advanced Study, FreemanDyson, who was one of the early figures in the study of random matrices1, announced thatthe pair correlation function Montgomery announced was identical to the one for the spacingsbetween eigenvalues of random Gaussian unitary matrices. Thus we have the final conjecturediscussed in the thesis:1Along with Eugene Wigner and othersCHAPTER 1. INTRODUCTION 12Conjecture 8 (Montgomery’s Pair Correlation Conjecture [13]). Let ζ(s) be the RiemannZeta function. The zeros of the Riemann Zeta function, normalized to have unit spacing onaverage, have pair correlation function:1−(sin(piu)piu)2. (1.21)As discussed in Chapter 4, this conjecture was already partially proven by Montgomeryin his paper introducing it. This fact strengthens the idea that the Hilbert-Po´lya Conjectureoffers a suitable approach to a proof or disproof of the Riemann Hypothesis.CHAPTER 1. INTRODUCTION 13Figure 1.1: Colour plot of ζ(s). Colour corresponds to Arg(ζ(s)). Note the trivial zeros onthe bottom, and the first 3 non-trivial zeros on the line <(s) = 12 . Also note that whilecircling around a small contour around each zero, every colour is encountered once. Figurecode from [7].Chapter2Riemann’s Wonderful FunctionI am definitely a mad man witha box.The Doctor2.1. IntroductionIn this chapter, we will expand further on the properties of ζ(s) and develop further intutionabout the Riemann zeros. We begin with an overview of prime-counting functions and willsee where ζ(s) arises naturally out of this theory. This will turn into a discussion of the J(x)and ψ(x) functions which will then end with a walkthrough of the theory behind the densityof the Riemann zeros. This will give enough background to sufficiently study the ideas laidout in Chapter 4.2.2. Prime Counting FunctionsLet us begin by making a few informal statements and then later making them more precise.First, take the logarithm of both sides in the result of Theorem 3. This yields:log ζ(s) =∑p(∑np−nsn)(2.1)14CHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 15after noting the series expansion log(1−x) = −x− 12x2− 13x3−. . . , we can define the functionJ(x) =12[∑pn<x1n+∑pn≤x1n](2.2)and apply it as a Stieltjes measure for Equation 2.1. Recall the definition of Riemann-Stieltjesintegration.Definition 9 (Riemann-Stieltjes Integration (Definition 6.2 in [17])). Let [a, b] be an inte-gration domain with partition P , such that P = {a = x0 < x1 < · · · < xn = b}, without lossof generality. The Riemann-Stieltjes sum is defined asS(P, f, g) =n−1∑i=0f(ci)[g(xi+1)− g(xi)] (2.3)for some ci in the interval [xi, xi+1], and for some real functions f, g. The Riemann-Stieltjesintegral∫ ba f(x)dg(x) equals A if for every  > 0, there exists a δ such that max(|xi+1−xi|) < δimplies that |S(P, f, g)− A| <  for all choices of ci.Using J(x) as our g function, and taking b→∞ gives the integrallog ζ(s) = s∫ ∞0x−1−sJ(x)dx. (2.4)Von Mangoldt realized that this could be written asζ ′(s)ζ(s)= −∫ ∞0x−s log(x)dJ(x). (2.5)Here, log(x)dJ(x) is a measure that weights log pnn to prime powers pn. We can write it as aStieltjes measure ψ(x):ψ(x) =∑pn<xlog(p). (2.6)this means that via integrating Equation 2.5 by parts we get−ζ ′(s)ζ(s)= s∫ ∞0x−s−1dψ(x), (2.7)CHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 16since the boundary terms are zero. In the next few subsections, we would like to give amore precise proof of these statements. First, we will take a tangent and talk about Perron’sFormula and Mellin Transforms, which will prove to be useful.2.2.1. Perron’s Formula and Mellin Transform TheoryPerron’s Formula is an expression for the infinite sum of arithmetic functions. It is given interms of the inverse of what will be subsequently defined as a Mellin transform.Theorem 10 (Perron’s Formula (Theorem 11.18 in [2])). Let {a(n)} be an arithmetic se-quence, and let h(s) be an absolutely convergent Dirichlet series for <(s) > c for somepositive c given byh(s) =∞∑n=1a(n)ns. (2.8)Let x > 0 be a real number. Then, Perron’s formula isA(x) =∑n≤xa(n) =12pii∫ d+i∞d−i∞h(z)xzzdz, (2.9)where d > c, and if x is an integer, a(x) is multiplied by a factor of 12 . Further, x > 0 forreal x.Proof. We defer the reader to page 245 of [2].Definition 11 (Mellin Transform). Let f be a real-valued function. The Mellin transformof f is given by{Mf}(s) = φ(s) =∫ ∞0xs−1f(x)dx. (2.10)The inverse of the Mellin transform is given by{M−1φ}(x) =12pii∫ c+i∞c−i∞x−sφ(s)ds. (2.11)The conditions under which one may take the inverse Mellin transform of a function φ(s)are given by the Mellin Inversion Theorem.Theorem 12 (Mellin Inversion). Let φ(s) be a complex-valued function. Suppose that φ(s)is analytic in the strip a < <(s) < b, and that φ(s) → 0 uniformly as =(s) → ±∞ forCHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 17any c ∈ (a, b). Further, suppose that integral of φ(s) on the line (c− i∞, c+ i∞) convergesabsolutely. Then, if{M−1φ}(x) =12pii∫ c+i∞c−i∞x−sφ(s)ds, (2.12)we have that{Mf}(s) = φ(s) =∫ ∞0xs−1f(x)dx. (2.13)Proof. We refer the reader to [14] for the proof, as it does not offer any useful insights ormethodology.We can thus see that Equation 2.9 in Theorem 10 is given by an inverse Mellin transform.2.2.2. First Proof of ψ(x) FormulaIn this subsection, we will give a proof of the formula for ψ(x). This will provide usefulbackground when discussing the density of the Riemann zeros. First, let us define thelogarithmic integral.Definition 13 (Logarithmic Integral). Let x be a real variable greater than or equal to 1.The logarithmic integral function is defined asLi(x) = lim→0[∫ 1−0dtlog t+∫ x1+dtlog t](2.14)Riemann proved the following theorem, which will require a proof of an important inte-gration formula in its own proof.Theorem 14 (Logarithmic Integral Theorem). Let a be real and greater than 1. Then,Li(x) =12pii1log x∫ a+i∞a−i∞dds[log(s− 1)s]xsds. (2.15)Before we prove Theorem 14, let us first prove the following.Theorem 15 (An Integration Formula). Let a be real and greater than 1. Further, leta > <(β). Then,12pii∫ a+i∞a−i∞1s− βysds ={ yβ if y > 112yβ if y = 10 if 0 < y < 1. (2.16)CHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 18Proof. Following the proof from Section 1.14 of [8], consider the following integral for <(s−β) > 0. Applying the Fourier Inversion Theorem for a > <(β)1s− β=∫ ∞1x−sxβ−1dx. (2.17)1a+ iµ− β=∫ ∞0e−iλµeλ(β−α)dλ (2.18)we get∫ ∞−∞1a+ iµ− βeiβxdµ ={2piex(β−a) If x > 00 If x < 0. (2.19)Assuming that a > <(β), taking the integral from ±i∞ offset by the positive real numbera, we get the result:12pii∫ a+i∞a−i∞1s− βysds ={ yβ if y > 112yβ if y = 10 if 0 < y < 1. (2.20)Next, let us consider, without proof, some results for the exact value of J(x). VonMangoldt proved Riemann’s original formula for J(x):J(x) = Li(x)−∑ρLi(xρ)− log 2 +∫ ∞xdtt(t2 − 1) log t, (2.21)where ρ corresponds to the Riemann zeros. We would like to use this result to relate J(x) toψ(x) in a more rigorous way than considered in the introduction of this chapter. This resultgives the Stieltjes measure for x > 1:dJ =(1log x−∑ρxρ−1log x−1x(x2 − 1) log x)dx (2.22)Then, informally we can say dJ log x = dψ:dψ =(1−∑ρxρ−1 −∑ρx−2n−1)dx. (2.23)CHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 19This led Von Mangoldt to guessψ = x−∑ρxρρ+∑nx−2n2n+ C (2.24)as the formula for ψ(x). We now begin the derivation of this claim. First, define the Πfunction.Proof. Following Chapter 3 in [8]:Definition 16 (The (Capital) Pi Function). Let Γ(z) be defined as before. Then,Π(z) = Γ(z + 1) = zΓ(z) =∫ ∞0e−ttzdt. (2.25)Recall that we have as in Equation 2.7−ζ ′(s)ζ(s)= s∫ ∞0ψ(x)x−s−1dx. (2.26)Using a now-well-defined Mellin transform, we getψ(x) =12pii∫ a+i∞a−i∞[−ζ ′(s)ζ(s)]xsdss. (2.27)Now lets define the Λ function as the weight assigned to n by the dψ measure:Λ(n) ={ log p if n = pα0 else(2.28)Then, we can rewrite Equation 2.26 as−ζ ′(s)ζ(s)=∞∑n=2Λ(n)ns. (2.29)Inserting Equation 2.29 into Equation 2.27 yields:ψ(x) =12pii∫ a+i∞a−i∞[∞∑n=2Λ(n)ns]xss(2.30)CHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 20ψ(x) =∞∑n=2Λ(n)2pii∫ a+i∞a−i∞(xn)s dss.Theorem 17 (Term-by-term integration (Section 9.9 of [1])).Suppose that uk is integrable on [a, b] for all k ≥ 1 and that∑∞k=1 converges uniformlyon [a, b]. Then, the limit function f(x) is integrable on [a, b] and∫ baf(x)dx =∞∑k=1∫ bauk(x)Noting that Equation 2.30 converges uniformly, by Theorem 17 it follows thatψ(x) =∑n<xΛ(n), (2.31)using the integration formula proven in Theorem Second Proof of ψ(x) FormulaPartially following Section 3.2 in [8], we can use the symmetry of ζ(s) and ξ(s) to writeΠ(s2)pis2 (s− 1)ζ(s) = ξ(0)∏p(1−sp). (2.32)Taking the logarithmic derivative will giveddslog Π(s2)−12log pi +1s− 1+ζ ′(s)ζ(s)=∑p11− sp(−1p)(2.33)Next, solve for − ζ′(s)ζ(s) and substituteΠ(s) =∞∏n=1(1 +sn)−1(1 +1n)s(2.34)CHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 21to get−ζ ′(s)ζ(s)=1s− 1−∑p1s− p+∞∑n=1[−1s+ 2n+12log(1 +1n)]−log pi2, (2.35)which gives us−ζ ′(0)ζ(0)= −1−∑p−1p+∞∑n=1[−12n+12log(1 +1n)]−log pi2. (2.36)Using equation 2.36 and equation 2.35, we get−ζ ′(s)ζ(s)+ζ ′(0)ζ(0)=[1 +1s− 1]−∑p[1p+1s− p]−∞∑n=1[1s+ 2n−12n]. (2.37)Simplifying this will give us−ζ ′(s)ζ(s)=ss− 1−∑psp(s− p)+∞∑n=1s2n(s+ 2n)−ζ ′(0)ζ(0)(2.38)Plug equation 2.38 into equation 2.27 to getψ(x) =12pii∫ a+i∞a−i∞[−ζ ′(s)ζ(s)]xsdss(2.39)=12pii∫ a+i∞a−i∞x2s− 1ds−∑p∫ a+i∞a−i∞xsp(s− p)ds+12pii[∫ a+i∞a−i∞(∑nxs2n(s+ 2n)+[−ζ ′(0)ζ(0)]1s)ds]Let t = s− β and use the integration formula proven in Theorem 15 which then gives12pii∫ a+i∞a−i∞1s− βysds ={ yβ if y > 10 if y < 1(2.40)and12pii∫ a+i∞a−i∞xss− β=12pii∫ a+i∞−βa−i∞−βxβxtdtt=xβ2pii∫ <(a−β)+i∞<(a−β)−i∞xttdt = xβCHAPTER 2. RIEMANN’S WONDERFUL FUNCTION 222.3. Density of the Non-Trivial RootsLet N denote the number of roots a function f has in a complex contour γ, and let P bethe number of poles in this contour. Then by Cauchy’s Argument Principle (Theorem 2 inSection 3.1 of [9]),N − P =12pii∮f ′(z)f(z)dz. (2.41)Consider the complex rectangle RT 0 < =(s) < T and 0 < <(s) < 1, with T > 0. Then,integrating around RT gives:N(T ) =12pii∫RTζ ′(s)ζ(s)ds (2.42)Theorem 18. Let RT be as before with T > 0. Then,N(T ) =T2pilog(T2pi)−T2pi+O(log T ) (2.43)In particular, we are interested in Backlund’s estimate (page 31 of [6]):∣∣∣∣N(T )−(T2pilogT2pi−T2pi+78)∣∣∣∣ < 0.137 log T + 0.443 log log T + 4.350 (2.44)for T > 0. We leave these results without proof, however the proof is readily available inSection 6.7 of [8].Chapter3A Primer on PhysicsPhysics is like sex: sure, it maygive some practical results, butthat’s not why we do it.Richard P. Feynman3.1. IntroductionIn this section we outline some of the main principles of physics, and in particular quantummechanics so that the reader may have sufficient background to understand the terminologyin the discussion of Berry’s Hilbert-Po´lya operator. Due to the non-rigorous nature of physicsas a discipline, some of the following is non-rigorous. The author, caught in the crossfirebetween the two disciplines, tries his best to introduce rigour whenever possible.3.2. Classical MechanicsClassical mechanics is the study of particle motion with the assumption that the motionis represented by a continuous function. Here, a ‘particle’ is any object whose volumecan be neglected. Furthermore, the various observables (mathematical quantities that anexperimentalist can measure) are also given by continuous functions, such as energy, position,and momentum. In contrast, quantum mechanics ‘discretizes’ the values of energy a particlecan have. All material in this section can be found in [10].23CHAPTER 3. A PRIMER ON PHYSICS 243.2.1. Lagrangian FormalismThe Lagrangian formalism is the standard varational calculus method for determining the‘equations of motion’ of a system. These equations of motion are usually in the form of asecond-order differential equation in position. The most familiar of these should be Newton’sSecond Law:Definition 19 (Newton’s Second Law). The equations of motion of a particle with mass mare of the formmd2xdt2= f(x,dxdt)(3.1)for some continuous and differentiable function f .The Lagrangian formalism begins with the definition of the Lagrangian:Definition 20 (The Lagrangian). The Lagrangian is a function of variables q and q˙ givenbyL(q, q˙) = T − V, (3.2)where T is the kinetic energy function, V the potential energy, and the dot referring to atime derivative.In this definition, q and q˙ = ddtq are some arbitrary variables that characterize the system.Typically we take q to be position (of say, a particle), and q˙ to be velocity. Before wedefine the functional action, we have to summarize the Fundamental Lemma of Calculus ofVariations.Definition 21 (Differentiability Classes). A function f is said to be of differentiability classCk if f is k times continuously differentiable.Lemma 22 (Fundamental Lemma of Calculus of Variations). Let f be a k-times continuouslydifferentiable function (of class Ck) on [a, b]. If for every function g on [a, b] of class Ckwith g(a) = g(b) = 0 we have that∫ baf(x)g(x)dx = 0, (3.3)then f(x) = 0 on [a, b].CHAPTER 3. A PRIMER ON PHYSICS 25Now, we can define the action and examine some of its properties.Definition 23 (The Action). The action is given by the functionalS[L] =∫ t1t0L(q, q˙)dt. (3.4)The action is an integral over all ‘paths’ parameterized by L, and particularly q(t). q(t) isnormally associated with position, so a ‘path’ here just means some trajectory of a particle.By assumption, q(t) has continuous first derivatives. We will identify some form of q(t) asthe g(x) function in Lemma 22. We wish to find an extremal value of this action in orderto derive the equations of motion. In order to do so, we assume what physicists call the‘Principle of Least Action’. This essentially means that at the boundary of the integral,namely at t0, t1, we assume that the variation of the action δS, and its variables δq, δq˙ arezero. We can then take the variation of the action:δS = 0 =∫ t1t0(∂L∂qδq +∂L∂q˙δq˙)dt. (3.5)Integrating the ∂L∂q˙ δq˙ term by parts yields:δS = 0 =∫ t1t0(∂L∂qδq +ddt(∂L∂q˙)δq)dt+∂Ldqδq∣∣∣∣t1t0. (3.6)The boundary term is zero by the Principle of Least action. Since the variation of the actionis zero, the integrand is also zero, and within the range (t0, t1), we have that δq > 0 whichimplies that by Lemma 22∂L∂q−ddt∂L∂q˙= 0. (3.7)This is the coveted Euler-Lagrange equation. Furthermore, this definition is easily general-izable to multiple q-variables. The general Euler-Lagrange equations are:Definition 24 (The Euler-Lagrange Equations). Let the Lagrangian L be defined as before.The Euler-Lagrange equations are given by:n∑i=1[∂L∂qi−ddt(∂L∂q˙i)]= 0. (3.8)CHAPTER 3. A PRIMER ON PHYSICS 263.2.2. Hamiltonian FormalismThe Hamiltonian formalism is equivalent to the Lagrangian formalism under a transformationof coordinates. Roughly speaking, one transforms the velocity coordinate into momentum.This simplifies the transition from classical physics to quantum physics immensely, becausemomentum is a ‘conserved’ quantity whereas velocity is not. A conserved quantity is onethat is constant in time. We begin by defining the Hamiltonian:Definition 25 (The Hamiltonian). The Hamiltonian H is a function of variables q and p,where p and its time derivative are given byp =∂L∂q˙p˙ =∂L∂q.(3.9)The Hamiltonian is then defined asH =∂L∂q˙q˙ − L. (3.10)Taking a variation of the Hamiltonian gives the following:δH =∂H∂qδq +∂H∂q˙δq˙ (3.11)δH = q˙δp+ pδq˙ −∂L∂qδq −∂L∂pδp˙ (3.12)Matching coefficients from Equations 3.11 and 3.12 gives Hamilton’s equations.Definition 26 (Hamilton’s Equations). Let H denote the Hamiltonian. Hamilton’s equationsare given by:∂H∂q= −p˙∂H∂p= q˙.(3.13)We can write these equations in a more concise form using Poisson brackets.CHAPTER 3. A PRIMER ON PHYSICS 27Definition 27 (Poisson Brackets). Let H be the Hamiltonian, and let f be some continuousand differentiable function of q, p. Then the Poisson brackets of f,H with respect to q, p aregiven by{f,H}q,p =∂f∂q∂H∂p−∂f∂p∂H∂q(3.14)We can then write Hamilton’s equations with some function f as:dfdt= {f,H}q,p +∂f∂t. (3.15)Admittedly, doing this is rather unmotivated. Once we get to the quantum formalism ofphysics, we shall see that we can make a formal analogy to this equation by identifying thePoisson brackets to a commutation relation, and the function f to some infinite-dimensional,self-adjoint and Hermitian operator.3.3. Quantum MechanicsIn this section, we outline the relevant components of quantum mechanics. We begin witha rigorous treatment of Hilbert spaces, and determine a matrix representation of infinite-dimensional, self-adjoint and Hermitian operators. We then outline how this mathematicalformalism is used in the study of quantum mechanics. Again, the purpose of this section isto familiarize the reader with the language of quantum mechanics that will be used in latersections. For reference, one may read [11].3.3.1. Hilbert SpacesA Hilbert space is a generalization of a vector space. Given two elements x, y in someHilbert space H with inner product 〈x|y〉, we have the following axioms. Note that thefollowing notation convention is set by physicists, and is backwards in axioms 2 and 4 fromthe convention set by mathematicians. If you are unfamiliar with either convention, this isprobably a good thing. Bars denote complex conjugates.Definition 28 (Axioms of Hilbert Spaces).1. 〈x|y〉 = 〈y|x〉2. 〈ax1 + bx2|y〉 = a〈x1|y〉+ b〈x2|y〉, where a, b ∈ C.CHAPTER 3. A PRIMER ON PHYSICS 283. 〈x|x〉 ≥ 0.4. 〈x|ay1 + by2〉 = a〈x|y1〉+ b〈x|y2〉, where a, b ∈ C.5. H is separable.6. {|ψn〉} is complete, so we can write||ψ||2 =∞∑n=1|〈φn|ψ〉|2. (3.16)From Axiom 5, we have that there exists a countable set A of vectors dense in H. Forexample, letA = {|ψ1〉+ |ψ2〉+ · · ·+ |ψk〉+ . . . }.Take out every vector |ψk〉, which can be a linear combination of |ψ1〉, . . . , |ψk−1〉. Then, weobtain an infinite set of linearly independent vectorsB = {|φ1〉+ |φ2〉+ · · ·+ |φn〉+ . . . } ⊂ A. (3.17)By construction, B is dense and can be made orthogonal by the Gram-Schmidt orthogonal-ization process such that〈φn|φm〉 = δnm ={1 m = n0 elseThus, for all |ψi〉 in H, we have:|ψi〉 =∞∑n=1|φn〉an. (3.18)Thus, we have〈φm|ψ〉 =∞∑n=1〈φm|φn〉an =∞∑n=1δnman = am,giving the result|ψ〉 =∞∑n=1|φn〉〈φn|ψ〉. (3.19)CHAPTER 3. A PRIMER ON PHYSICS 29From Axiom 6, it follows that〈ψ|ψ〉 = 〈ψ|1|ψ〉 =∞∑n=1〈ψ|φn〉〈φn|ψ〉. (3.20)We have thus found a diagonal representation of the identity in this Hilbert space:1 =∞∑n=1|φn〉〈φn| (3.21)Now, we can define the norm and metric.Definition 29 (Norm). The norm of x is given by||x|| =√〈x|x〉. (3.22)Definition 30 (Metric). The metric is given byd(x, y) = ||x− y|| =√〈x− y|x− y〉. (3.23)3.3.2. Operators in Quantum MechanicsAn operator is a mapping between vector spaces. In this thesis, we will be consideringoperators as mappings between infinite-dimensional Hilbert spaces. First, some elementarydefinitions.Definition 31 (Transpose Operation). Let A be an n×m matrix. At, called A-transpose isdefined such thatAtij = Aji. (3.24)This definition can be readily extended to infinite dimensions, however not all infinite-dimensional matrices have well-defined transposes. For the purposes of quantum mechanics,one always works with matrices that do have well-defined transposes.Definition 32 (Hermitivity). An operator A is Hermitian if A† = A, where the ‘†’ symbolrepresents the conjugate transpose operation. In other words,〈Aψ|φ〉 = 〈ψ|A|φ〉CHAPTER 3. A PRIMER ON PHYSICS 30Definition 33 (Self-Adjointivity). An operator A is self-adjoint if:〈A†ψ|φ〉 = 〈ψ|A|φ〉. (3.25)For an infinite-dimensional Hilbert space, we have that if A is Hermitian and self-adjoint,(A†)† = A. In quantum mechanics, all physical observables (energy, position, momentum,etc.) are Hermitian and self-adjoint operators in some infinite-dimensional Hilbert space.We now define some useful quantities.Definition 34 (Expectation Value). For some vector ψ with norm ||ψ|| = 1, the expectationvalue of an operator A with respect to ψ is given by〈A〉ψ =〈ψ|A|ψ〉〈ψ|ψ〉= 〈ψ|A|ψ〉. (3.26)As this is a measurable quantity by an experimenter, we require this quantity to be real,hence the restriction that A be Hermitian and self-adjoint.Definition 35 (Eigenvalue of an Operator). IfA|ψ〉 = a|ψ〉, (3.27)where a ∈ C, then a is called the eigenvalue of A. For a given vector |ψ〉, if a exists, a isunique.3.3.3. Matrix Representation of an OperatorSuppose we have an operator acting on a vector to produce another vector: A|ψ〉 = |ψ′〉.Then,A(1)|ψ〉 = |ψ′〉 = A(∞∑n=1|φn〉〈φn|)|ψ〉.Multiplying both sides on the left by 〈φm| gives:〈φm|A(∞∑n=1|φn〉〈φn|)|ψ〉 =∞∑n=1〈φm|A|φn〉〈φn|ψ〉 = 〈φm|ψ′〉.CHAPTER 3. A PRIMER ON PHYSICS 31Recall that the expansion of a vector |ψ〉 is given by:|ψ〉 =∞∑n=1|φn〉〈φn|ψ〉. (3.28)Hence, the expansion coefficients an in Equation 3.18 are given by:〈φm|ψ′〉 (3.29)with respect to the orthnormal basis {φm}. Therefore, the matrix elements of an operatorrepresentation of A are:Amn = 〈φm|A|φn〉. (3.30)Further, we can write a vector as:|ψ〉 →〈φ1|ψ〉〈φ2|ψ〉...〈φn|ψ〉...=a1a2...an...(3.31)3.3.4. Fundamental EquationsWe have stated that in quantum mechanics, observables such as position, momentum andenergy are given by Hermitian, self-adjoint operators. Further, the quantum state of aparticle is given by some vector |ψ〉. The time-dependence on both of these quantities aregiven by differential equations.Definition 36 (The Schro¨dinger Equation). Let H be the Hermitian and self-adjoint opera-tor in an infinite-dimensional Hilbert space corresponding to the classical Hamiltonian, andlet |ψ〉 be some vector in this space. The time dependence of |ψ〉 is given by the Schrodingerequation:H|ψ(t)〉 = i~∂∂t|ψ(t)〉 (3.32)where i2 = −1 and ~ is a numerical constant with units of [energy][time].Definition 37 (Operator Time Dependence). Let A be a Hermitian and self-adjoint operatorCHAPTER 3. A PRIMER ON PHYSICS 32in an infinite-dimensional Hilbert space, and let H be the Hamiltonian operator as before.The time dependence of A is given byddtA =i~[H,A] +∂A∂t(3.33)where [H,A] is the commutation relation [H,A] = HA− AH.Note that this equation is directly analogous to the classical equation for the time de-pendence of a continuous function f as in Equation 3.15, if we identify the Poisson bracketto the commutator, and f to the operator A.Chapter4The Hilbert-Po´lya ConjectureI am and always will be theoptimist. The hoper of far-flunghopes and the dreamer ofimprobable dreams.The Doctor4.1. Introductory NotesThe Hilbert-Po´lya Conjecture offers a promising methodology in regards to proving the Rie-mann Hypothesis. The fields of linear operators, matrices and linear algebra are arguablymore well-understood than the interface between complex analysis and analytic numbertheory. For one, the calculations themselves are somewhat simpler. Apart from this, theconjecture also points to connections between prime number theory, and the study of quan-tum mechanics.4.2. Random Matrix TheoryRandom matrices were first studied by Wigner and later by Dyson as a model for the energylevels of atomic nuclei. There are three main types of random matrices, namely Gaussianorthogonal, unitary and symplectic. We will be interested mainly in the Gaussian unitarymatrices. All of the following can be found in Chapter 6 of [12].33CHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 344.2.1. Gaussian Unitary MatricesA random Gaussian unitary matrix can be described by the joint probability distribution ofits eigenvalues. This is given for an N ×N matrix asPN(x1, . . . , xN) = CN exp(−N∑j=1x2j)∏j<k|xj − xk|2, (4.1)where CN is a normalization constant given by∫ ∞−∞. . .∫ ∞−∞PN(x1, . . . , xN)dx1 . . . dxN = 1 (4.2)which yields(CN)−1 = (2pi)N2 βN2 −βN(N−1)4 (Γ(1 +β2))−NN∏j=1Γ(1 +βj2) (4.3)4.2.2. Pair Correlation Function for Gaussian Unitary MatricesA pair correlation function of order n (otherwise known as an n-point correlation function)gives the probability of finding an eigenvalue of a random matrix around points x1, x2, . . . , xngiven no information about the location of any other eigenvalues. The function is defined asfollows.Definition 38 (The n-point Correlation Function). Let A be an N×N matrix. The n-pointcorrelation function is given byRn(x1, x2, . . . , xn) =N !(N − n)!∫ ∞−∞. . .∫ ∞−∞PN(x1, x2, . . . , xN)dxn+1 . . . dxN . (4.4)Note that the integrals are taken over all ni in between n and N .It is often more apt to work with the n-point cluster function Tn, defined in terms of then-point correlation function.Definition 39 (n-point Cluster Function). Let A be an N ×N matrix. The n-point clusterCHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 35function is defined asTn(x1, . . . , xn) =∑G(−1)n−m(m− 1)!M∏j=1RGj(xk, k ∈ Gj) (4.5)where G is any division of the integer indices (1, 2, 3, . . . , n) into m partitions (G1, G2, . . . ,Gm). For example,T1(x1) = R1(x)T2(x1, x2) = −R2(x1, x2) +R1(x1)R2(x2)T3(x1, x2, x3) = R3(x1, x2, x3)−R1(x1)R2(x2, x3)− · · ·+ 2R1(x1)R1(x2)R1(x3)(4.6)and so on.We can invert Equation 4.5 to express the correlation function in terms of the clusterfunction:Rn(x1, . . . , xn) =∑G(−1)m−nm∏j=1TGj(xk, k ∈ G). (4.7)By extending N →∞ and considering a ‘mean spacing’ α between eigenvalues of a randommatrix, we can define the mean-spaced cluster function.Definition 40 (Mean-Spaced n-point Cluster Function). Let N be defined as before, andlet α be the mean spacing between eigenvalues of an N × N matrix. Further, let Tn be then-point cluster function. The mean-spaced n-point cluster function is given asYn(y1, y2, . . . , yn) = limN→∞αnTn(x1, . . . , xn), (4.8)where yi = x1α .We now wish to find some of the values of these functions, in particular for the Gaussianunitary matrices. We will find that the values of these functions are eerily similar to thatof the statistics corresponding to the distribution of the imaginary parts of the non-trivialRiemann Zeta zeros. First, let us define the oscillator functions.Definition 41 (Oscillator Function). For integer j, the oscillator function is given byφj(x) = (2jj!√pi)−1/2 exp(x22)(−ddx)jexp (−x2). (4.9)CHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 36With some work, it is possible to show that for Gaussian unitary matrices, we can writethe mean-spaced 2-point cluster function in term of these oscillator functions asY2(x1, x2) = limN→∞(pi(2N)1/2N−1∑j=0φj(x1)φj(x2))2=(sin(pir)pir)2, (4.10)and the 2-point correlation function asR2(x1, x2) = 1−(sin(pir)pir)2. (4.11)The pair correlation function is plotted in Figure 4.1.0 2 4 6 8 100.920.940.960.981.00Figure 4.1: Plot of the pair correlation function Equation 4.11, with r as horizontal axis.4.3. Zeroes of ζ(s) and Random MatricesIn this section, we wish to describe the pair correlation of the imaginary parts of the non-trivial Riemann zeros, in particular show some evidence that it is the same function asEquation 4.11. This is the conjecture put forth by Montgomery in his seminal paper. Wethen consider a formal analogy between the spectrum (eigenvalues of the Hamiltonian) ofCHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 37a chaotic quantum system considered by Berry in [3] and the Riemann zeta zero countingfunction given by Von Mangoldt and Backlund earlier in the thesis. First, we begin byconsidering the distribution of Riemann zeros. The number of zeros in the critical strip withimaginary part less than or equal to E is given by the ‘Riemann staircase’.Definition 42 (Riemann Staircase).NR(E) =∞∑j=1Θ(E − Ej), (4.12)where Θ is the Heaviside step function, and not the Θ function related to Γ(s).As in Chapter 2, we see that there are estimates and known asymptotic behaviour forthis function. We resummarize a familiar theorem:Theorem 43 (Backlund’s Estimate for the Riemann Staircase.). Let ζ(s) denote the Rie-mann Zeta function, and let E denote the imaginary part of some complex number z in thecritical strip. Then, the number of zeros of ζ(s) with imaginary part less than or equal to Eis given by Equation 4.12. Furthermore, an approximation to this expression is given by〈NR(E)〉 =E2pi(log[E2pi]− 1)+78, (4.13)following the notation of [3].This is a slight modification to Theorem 18 to chapter 2. The spacing between eigenvaluesof random Gaussian unitary matrices have a similar form as the probability distribution P (S)of the normalized spacing between adjacent zeros Sj:Sj =(Ej+1 − Ej)〈dr(Ej+Ej+12)〉(4.14)where 〈Dr(E)〉 is the average density of zeros:〈Dr(E)〉 =ddE〈NR(E)〉 =12pilog(E2pi). (4.15)The Fourier Transform of the pair correlation function of the Riemann zeros is given byCHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 38Equation 9 in [3]:K(τ) = limM→∞{1MM∑j=1M∑k=1exp [2piiτ(xj − xk)]−sinMpiτpiτ}(4.16)where {xj} are the Riemann zeros scaled to have unit mean spacing. Montgomery’s conjec-ture in [13] can be restated as follows:Conjecture 44 (Montgomery’s Conjecture on the Fourier Transform of the Pair CorrelationFunction). Let K(τ) be the Fourier transform of the pair correlation function as defined inEquation 4.16. Further, note that the Fourier transform of Equation 4.11 is given byKGUE(τ) ={ |τ | for |τ | < 11 for |τ | > 1.(4.17)In [13], it is shown that K(τ) = KGUE(τ) for |τ | < 1. The Conjecture is that Equation 4.17holds for |τ | > 1.We now look at a chaotic quantum system described by Berry in [3]. This system followsthree properties:1. The Hamiltonian H has a classical limit. This means that the quantum-mechanicaloperator equations have a direct classical representation under Hamilton’s equations.2. The classical orbits given by Hamilton’s equations are chaotic (unstable)3. The classical orbits are not time-reversible (they are not symmetric under a coordinatechange t→ −t)Much of the following exposition will be schematic. Readers are forwarded to read [3]whenever there is ambiguity. First, we consider the analogue of Backlund’s zero countingfunction N for the real part E of the eigenvalues of the Hamiltonian of this quantum system.We work with a general Hamiltonian. We can separate N(E) into an ‘average’ part and a‘fluctuating’ (or oscillating) part:N(E) = 〈N(E)〉+Nosc(E). (4.18)CHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 39Then we get Equation 12 in [3]:Nosc(E) = N(E)− 〈N(E)〉 = =[∑p∞∑m=LBpm exp(iSpm(E)~)]. (4.19)The double sum is over orbits labelled with prime numbers p, and integers m traversals ofthat orbit. Spm is the action given by:Spm(E) =∮pµdqµ. (4.20)This action describes some chaotic trajectories obtained by Hamilton’s equations. Equation14 in [3] gives the amplitude coefficientBpm = 2pim sinh(mλp(E)2). (4.21)Then,Nosc =12pi∑p∞∑m=1sinh{mSp1~ }m sinh{mλp(E)2 }. (4.22)After some work, we can show this is equal toNosc(E) ≈1pi∑p∞∑m=11mexp(−mλp(E)2)sin(mSp1(E)~). (4.23)Unfortunately, the corresponding expression for ζ(s) is given by Equation 18 in [3]:NR,osc(E) ≈ −1pi∑p∞∑m=1sin(mE log p)mpp2, (4.24)if we identify the actions of the closed orbitsSpm = mR log pand the instability exponentλp = log p.CHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 40Implicitly, what we are suggesting is that the closed orbits are identified to prime numbersp. Unfortunately, now this equation equals Equation 4.22, but differs by a sign. We canstrengthen this analogy by considering the productP (E) =∞∏k=0ζ(12+ k − iE). (4.25)Theorem 45 (Convergence of P (E)).Let P (E) be defined as in Equation 4.25. Then,1. P (E) converges for real E.2. P (E) has the same zeros as ζ(12 + iE).Then, we get that the corresponding expression for NP,osc is given by a more immediatelyanalogous equation:NP,osc = −1pilimη→0= logP (E + iη) ≈ −12pi∑p∞∑m=1sinh{mE log p}m sinh{m log p2 }. (4.26)We can directly compare factors in the hyperbolic sines between this equation and withEquation 4.22. This concludes the schematic introduction to the formal analogy betweenthe statistics of the eigenvalues of the chaotic quantum system studied by Berry, and thestatistics of the distribution of the imaginary parts of the Riemann zeros. The oscillatingparts of the respective functions differ by a minus sign, which apart from lack of rigour, isa weakness in this approach. A slightly more rigorous discussion of this analogy is given byBerry and Keating in [4] and [5].4.4. The Berry-Keating OperatorMontgomery’s conjecture tantalizingly points a finger and says ‘hey, look over there’ to theidea that the Riemann zeros are somehow connected to the eigenvalues of a self-adjointunitary operator. We recall the Hilbert-Po`lya conjecture:Conjecture 46 (The Hilbert-Po´lya Conjecture). There exists a self-adjoint operator in aninfinite-dimensional Hilbert space whose eigenvalues are the non-trivial Riemann zeros.CHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 41Berry and Keating have speculated that a possible operator that may satisfy the Hilbert-Polya conjecture is given by the HamiltonianH =12(XP + PX) = −i(Xddx+12). (4.27)The eigenvectors of this Hamiltonian with eigenvalues E corresponding to functions ψ(X)are given fromHψ(X) = Eψ(X) (4.28)asψ(X) =AX1/2−iE. (4.29)Using a Fourier transform, we can compute the eigenfunction in momemtum space:φ(P ) =1√2pi∫ ∞−∞dXψ(X)e−iPX , (4.30)givingφ(P ) =A|P |1/2+iE2iEΓ(14 +12iE)14 −12iE=A√2pi|P/(2pi)|1/2+iEe2iθ(E) (4.31)where θ(E) is defined as in Equation 4.32:Θ(E) = arg Γ(14+E2)−E2log pi ≈E2logE2pi−E2−pi8. (4.32)Now, consider the following coordinate tranformation:X → X1 =2piP, P → P1 =XP 22pi. (4.33)We can then rewrite the transformed wavefunction ψ1 in terms of the untransformed mo-mentum wavefunction φ:ψ1(X1) =(2pi)1/4|X1|φ(√2pi|X1|). (4.34)Berry and Keating in [5] use this, Equation 4.30 and the functional equation for ζ(s) to writeCHAPTER 4. THE HILBERT-PO´LYA CONJECTURE 42the identity (Equation 6.12 in their paper)X1/2ζ(12− iE)ψE(X)− P1/2ζ(12+ iE)φE(P ) = 0. (4.35)Unfortunately, this approach fails at this last stage: the term on the right has a plus insidethe zeta function. If this were a minus sign, then this would be precisely the conditionrelating the Riemann zeros to the eigenvalues of the Hamiltonian in Equation 4.27.Chapter5Bibliography[1] T. M. Apostol, Mathematical analysis, (1974).[2] , Introduction to analytic number theory, Springer, 1976.[3] M. V. Berry, Riemann’s zeta function: A model for quantum chaos?, in Quantumchaos and statistical nuclear physics, Springer, 1986, pp. 1–17.[4] M. V. Berry and J. P. Keating, H= xp and the riemann zeros, in Supersymmetryand Trace Formulae, Springer, 1999, pp. 355–367.[5] , The riemann zeros and eigenvalue asymptotics, SIAM review, 41 (1999), pp. 236–266.[6] P. Borwein, The Riemann hypothesis: a resource for the afficionado and virtuosoalike, vol. 27, Springer, 2008.[7] B. Champion, Plot a complex function in mathematica, 2011.Retrieved from http://stackoverflow.com/questions/5385103/plot-a-complex-function-in-mathematica.[8] H. M. Edwards, Riemann’s zeta function, vol. 58, Academic Press, 1974.[9] S. D. Fisher, Complex variables, DoverPublications. com, 1990.[10] L. Landau, Mechanics: Volume 1 (course of theoretical physics) author: Ld landau,em lifshitz, publisher: Butterwor, (1976).43CHAPTER 5. BIBLIOGRAPHY 44[11] N. P. Landsman, 2006 lecture notes on hilbert spaces and quantum mechanics, 2013.Retrieved from http://www.math.kun.nl/~landsman/HSQM2006.pdf.[12] M. L. Mehta, Random matrices, vol. 142, Elsevier, 3 ed., 2004.[13] H. Montgomery, The pair correlation of zeros of the zeta function, in Proc. Symp.Pure Math, vol. 24, 1973, pp. 181–193.[14] A. D. Poularikas, Transforms and applications handbook, CRC Press, 2010.[15] B. Riemann, Ueber die anzahl der primzahlen unter einer gegebenen grosse, Ges. Math.Werke und Wissenschaftlicher Nachlaß, 2 (1859), pp. 145–155.[16] E. S. Rowland, Functional equations for the riemann zeta function and dirich-let l-functions, 2013. Retrieved from http://thales.math.uqam.ca/~rowland/papers/Functional_equations_for_the_Riemann_zeta_function_and_Dirichlet_L-functions.pdf.[17] W. Rudin, Principles of mathematical analysis, vol. 3, McGraw-Hill New York, 1964.[18] F. Topsok, Some bounds for the logarithmic function, Inequality theory and applica-tions, 4 (2006), p. 137.[19] R. M. (XKCD), Riemann-zeta, 2013. Retrieved from http://xkcd.com/113/.Appendices45AppendixAAppendixCode for Figure 1 (Mathematica). Reproduced from [7]Warning: This requires a decent (at time of writing, ignore this if Moore’s law has taken over)computer with lots of RAM and processing power (takes about 20 seconds with quad-coreIntelr i7r and 8GB of RAM). If reading from future, do not have lulz over this.Parametr icPlot [{ x , y} , {x , −22.5 , 3} , {y , −3, 30} ,P lotPo ints −> 200 , MaxRecursion −> 0 ,Mesh −> 65 , Co lorFunct ionSca l ing −> False ,ColorFunction −> (Hue [ Resca le [ Arg [ Zeta [# + I #2] ] ,{−Pi , Pi } , {0 , 1} + 0 . 5 ] , 1 ,Resca le [ Log [ Abs [ Zeta [# + I # 2 ] ] ] ,{− I n f i n i t y , I n f i n i t y } , {0 .1 ,2 } ] ] &) ,MeshFunctions −> {Log [Abs [ Zeta [#1 +I #2 ] ] ] &} ,Axes −> False ]46


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