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Zeros of the Dedekind Zeta-Function

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Zeros of the Dedekind Zeta-Function University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2019 Zeros of the Dedekind Zeta-Function Mashael Alsharif University of Mississippi Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Mathematics Commons Recommended Citation Alsharif, Mashael, "Zeros of the Dedekind Zeta-Function" (2019). Electronic Theses and Dissertations. 1542. https://egrove.olemiss.edu/etd/1542 This Thesis is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. ZEROS OF THE DEDEKIND ZETA-FUNCTION DISSERTATION A Dissertation presented in partial fulfillment of requirements for the degree of Master of Science in the Department of Mathematics The University of Mississippi by MASHAEL ALSHARIF December 2018 Copyright Mashael Alsharif 2018 ALL RIGHTS RESERVED ABSTRACT H. L. Montgomery proved a formula for sums over two sets of nontrivial zeros of the Riemann zeta-function. Assuming the Riemann Hypothesis, he used this formula and Fourier analysis to prove an estimate for the proportion of simple zeros of the Riemann zeta-function. We prove a generalization of his formula for the nontrivial zeros of the Dedekind zeta-function of a Galois number field, and use this formula and Fourier analysis to prove an estimate for the proportion of distinct zeros, assuming the Generalized Riemann Hypothesis. ii ACKNOWLEDGEMENTS Foremost, I would like to express my sincere gratitude to my advisor Prof. Micah Baruch Milinovich for the continuous support of my master study and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my master study. Besides my adviser, I would like to thank my family: my parents whose love, guidance, and support are with me in whatever I pursue. I wish to thank my loving and supportive husband, Fahad, and my wonderful son, Faisal, who provide unending inspiration. iii TABLE OF CONTENTS ABSTRACT ....................................... ii ACKNOWLEDGEMENTS ............................... iii 1 INTRODUCTION .................................. 1 1.1 Montgomery's Theorem. 3 1.2 Properties of the Dedekind zeta-function . 5 2 PROOF OF PROPOSITION 1.6 .......................... 11 3 PROOF OF COROLLARY 1.8 ........................... 15 4 PAIR CORRELATION FOR THE SELBERG CLASS ............... 20 5 PROOF OF THEOREM 1.7. ............................ 23 BIBLIOGRAPHY .................................... 28 VITA .......................................... 29 iv 1 INTRODUCTION For s 2 C, we let s = σ + it where σ; t 2 R. The Riemann zeta-function is initially defined as a Dirichlet series over the positive integers and an Euler product over the primes: 1 X Y ζ(s) = n−s = (1 − p−s)−1 (1.1) n=1 p prime for σ > 1. The equality, proved by Euler, follows from the Fundamental Theorem of Arith- metic. Riemann proved that ζ(s) can be continued analytically to Cnf1g with a simple pole at s = 1. Riemann also proved that ζ(s) satisfies the functional equation s s 1 1 − 2 − 2 (1−s) π Γ( 2 )ζ(s) = π Γ( 2 )(1 − s))ζ(1 − s): (1.2) From the poles of Γ(s) at s = 0; −1; −2; −3;:::, he observed that ζ(s) has simple zeros at s = −2; −4; −6; ··· . These are called the trivial zeros of the zeta function. He further noted that ζ(s) has infinitely many zeros in the critical strip, 0 ≤ σ ≤ 1, which are known as the non-trivial zeros of ζ(s). We denote the nontrivial zeros of ζ(s) as ρ = β + iγ. From the functional equation, if ρ is a nontrivial zero then so is 1 − ρ. Since ζ(s) = ζ(¯s), if ρ is a nontrivial zero then so isρ ¯. From this, Riemann observed that the zeros are symmetric 1 about the real axis and about the line σ = 2 . He made the following famous conjecture. Riemann Hypothesis. All nontrivial zeros of ζ(s) in the critical strip are on the critical 1 line σ = 2 . 1 Riemann introduced the zeta function as tool to study the prime numbers. Logarith- mically differentiating the Euler product for ζ(s) we have 1 d ζ0 X Λ(n) log ζ(s) = (s) = − ds ζ ns n=1 for σ > 1, where the von Mangoldt function Λ(n) = log p if n = pk for a prime p and k 2 N and Λ(n) = 0 otherwise. Using Riemann's ideas, in 1896, Hadamard and de la Vall´eePoussin independently proved that X Λ(n) ∼ x n≤x as x ! 1 by carefully studying the zeros of ζ(s). The key is to show that ζ(s) has no ζ0 nontrivial zeros on the line σ = 1 (so that ζ (s) has no poles on the line σ = 1). This asymptotic formula is equivalent to: Theorem 1.1 (Prime Number Theorem). As x ! 1, we have X x 1 ∼ log x p≤x where the sum runs over the primes p. This theorem was originally conjectured by Gauss and Legendre. The properties of ζ(s) described above and the history of the Prime Number Theorem can be found in Davenport's book [Dav00]. Much effort has gone into studying the nontrivial zeros ρ = β +iγ of ζ(s). It is known that X T T T N(T ) := 1 = log − + O(log T ) (1.3) 2π 2π 2π 0<γ≤T as T ! 1. This was conjectured by Riemann and proved by von Mangoldt [Dav00]. Here the zeros are counted with multiplicity meaning that a zero with multiplicity m is counted m times in the sum. In a now famous paper, Montgomery [Mon73] studied the pair correlation 2 of the zeros of ζ(s) assuming the Riemann Hypothesis. We now describe Montgomery's theorem and a corollary. The main goal of my thesis will be to generalize Montgomery's results to the Dedekind zeta-function of a Galois number field. 1.1 Montgomery's Theorem. We assume the Riemann Hypothesis in this section so that the nontrivial zeros can 1 be written ρ = 2 + iγ. From (1.3), the average spacing between consecutive γ 2 (0;T ] is length(0;T ] T 2π ≈ ≈ = # γ 2 (0;T ] T log T log T 2π n log T o as T ! 1. So the sequence γ 2π has average spacing equal to 1 as T ! 1. With this in mind, Montgomery was interested in studying sums like X log T R (γ − γ0) 2π 0<γ,γ0≤T where γ and γ0 run over the imaginary parts of two sets of nontrivial zeros of ζ(s). Montgomery defined the function 2π X 0 4 F (α) = F (α; T ) = T iα(γ−γ )w(γ − γ0); where w(u) = ; T log T 4 + u2 0<γ,γ0≤T where α and T ≥ 2 are real. Here γ and γ0 run over the imaginary parts of two sets of nontrivial zeros of ζ(s). He was interested in this function because, for R; R^ 2 L1(R), one can show that X log T T log T R (γ − γ0) w(γ − γ0) = F (α) R^(α)dα: (1.4) 2π 2π ˆ 0<γ,γ0≤T R 3 Here R^ is the Fourier transform of R defined as R^(α) = R(u) e−2πiuαdu: ˆR Using the definition of the Fourier transform, we will prove the analogue of (1.4) for the zeros of the Dedekind zeta-function in Chapter 2 and the same proof can be used to prove (1.4). Montgomery [Mon73] proved the following theorem about the function F (α). Theorem 1.2. Assume the Riemann Hypothesis. For real α and T ≥ 0, we have that F (α) is real, F (α) ≥ 0, and F (−α) = −F (α): For α 2 [0; 1] we have F (α) = 1 + o(1) T −2α log T + α + o(1) as T ! 1. We will prove an analogue of this theorem for the zeros of the Dedekind zeta-function. Originally Montgomery proved this theorem for α 2 (0; 1) but it was later extended to α 2 [0; 1] by Goldston and Montgomery [GM87]. Julia Mueller [Mue83] was the first to observe that F (α) ≥ 0 for all α 2 R. We will give a modification of her proof for the zeros of the Dedekind zeta-function in Chapter 2. The importance of Montgomery's theorem is that we can now estimate the right-hand side of (1.4) for a function R 2 L1(R) with supp(R^) ⊆ [−1; 1]. With these conditions, for most nice functions R, Montgomery's theorem implies that X log T T log T 1 R (γ − γ0) w(γ − γ0) = R^(0) + jαjR^(α)dα + o(1) : (1.5) 2π 2π ˆ 0<γ,γ0≤T −1 We will prove a similar formula for the zeros the Dedekind zeta-function. 4 We state one of the several important corollaries that Montgomery derived from his theorem and (1.5). Let s 1 N (T ) = # 0 < γ ≤ T : ρ = 2 + iγ is a simple zero of ζ(s) : Choosing the Fourier pair sin πu2 R(u) = ; R^(α) = max(1 − jαj; 0) πu in (1.5), he proved the following estimate for N s(T ) which shows that asymptotically at least two thirds of the nontrivial zeros of ζ(s) are simple. Corollary 1.3. Assume the Riemann Hypothesis. Then 2 N s(T ) ≥ + o(1) N(T ) 3 as T ! 1. It was later observed by Montgomery and Taylor [Mon75] and Cheer and Goldston [CG93] that the constant 2=3 can be very slightly improved using a more complicated choice of Fourier pair R and R^.
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