DOCSLIB.ORG

A Note on the Zeros of Zeta and L-Functions 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Note on the Zeros of Zeta and L-Functions 1 A NOTE ON THE ZEROS OF ZETA AND L-FUNCTIONS EMANUEL CARNEIRO, VORRAPAN CHANDEE AND MICAH B. MILINOVICH 1 Abstract. Let πS(t) denote the argument of the Riemann zeta-function at the point s = 2 + it. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound (and prove some related results) for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. 1. Introduction Let ζ(s) denote the Riemann zeta-function and, if t does not correspond to an ordinate of a zero of ζ(s), let 1 S(t) = arg ζ( 1 +it); π 2 where the argument is obtained by continuous variation along straight line segments joining the points 1 2; 2 + it, and 2 + it, starting with the value zero. If t does correspond to an ordinate of a zero of ζ(s), set 1 S(t) = lim S(t+") + S(t−") : 2 "!0 The function S(t) arises naturally when studying the distribution of the nontrivial zeros of the Riemann zeta-function. For t > 0, let N(t) denote the number of zeros ρ = β + iγ of ζ(s) with ordinates satisfying 1 0 < γ ≤ t where any zeros with γ = t are counted with weight 2 . Then, for t ≥ 1, it is known that t t t 7 1 N(t) = log − + + S(t) + O : 2π 2π 2π 8 t In 1924, assuming the Riemann hypothesis, Littlewood [12] proved that log t S(t) = O log log t as t ! 1. The order of magnitude of this estimate has never been improved. Using the Guinand-Weil explicit formula for the zeros of ζ(s) and the classical Beurling-Selberg majorants and minorants for characteristic functions of intervals, Goldston and Gonek [9] established Littlewood's estimate in the form 1 log t jS(t)j ≤ +o(1) (1.1) 2 log log t as t ! 1. The authors [4] recently sharpened this inequality and proved the following theorem. Theorem 1. Assume the Riemann hypothesis. Then 1 log t jS(t)j ≤ +o(1) 4 log log t for sufficiently large t, where the term of o(1) is O(log log log t= log log t). 2000 Mathematics Subject Classification. 11M06, 11M26, 11M36, 11M41, 41A30. Key words and phrases. Riemann zeta-function, automorphic L-functions, Beurling-Selberg extremal problem, extremal functions, exponential type. 1 In this note, we give a new and much simpler proof of Theorem 1. As in the work of Goldston and Gonek mentioned above, we use the explicit formula in conjunction with the classical majorants and minorants of exponential type for characteristic functions of intervals that were constructed by Beurling and Selberg. In [4], two proofs of Theorem 1 are given. The more direct of these proofs proceeds as follows. Assuming the Riemann hypothesis, for t not corresponding to an ordinate of a zero of ζ(s), it is shown in [4] that 1 X S(t) = F (t−γ) + O(1); π γ where 1 x F (x) = arctan − ; (1.2) x 1 + x2 1 and the sum runs over the nontrivial zeros ρ = 2 + iγ of ζ(s) counted with multiplicity. Drawing upon the work of Carneiro and Littmann [5], upper and lower bounds for S(t) are established by replacing F (x) in the sum over zeros by (optimally chosen) real entire majorants and minorants of F (x) and then applying the explicit formula. Using similar ideas, Chandee and Soundararajan [6] proved that log 2 log t log jζ( 1 +it)j ≤ +o(1) 2 2 log log t as t ! 1, also under the assumption of the Riemann hypothesis. The theory of extremal functions of exponential type has also been recently used in [3] to improve the upper and lower bounds for the pair correlation of zeros of ζ(s) under the Riemann hypothesis. In Section 4, we discuss how to extend Theorem 1 to a large class of L-functions and prove a number of other related results. If an L-function is self-dual, our new proof of Theorem 1 generalizes in a natural way. However, if an L-function is not self-dual, then our new proof does not generalize but we show that an analogue of Theorem 1 is still possible using a modification of our previous approach in [4]. 2. Beurling-Selberg majorants and minorants For functions R 2 L1(R), let Z 1 Rb(ξ) = R(x) e−2πixξ dx −∞ denote the Fourier transform of R. In the 1930s, Beurling observed that the real entire functions 2 ( 1 ) 2 sin πz X sgn(m) 2 sin πz H±(z) = + ± π (z−m)2 z πz m=−∞ satisfy the inequalities H−(x) ≤ sgn(x) ≤ H+(x) for all real x, have Fourier transforms Hb ±(ξ) supported in [−1; 1], and satisfy Z 1 ± H (x)−sgn(x) dx = 1: −∞ Moreover, Beurling showed that among all real entire majorants and minorants for sgn(x) with Fourier transforms supported in [−1; 1], the functions H±(x) minimize the L1(R)-distance to sgn(x). The details of this construction can be found in [20]. For t > 0, let χ[−t;t](x) denote the characteristic function of the interval [−t; t] (normalized at the end- points). From the above properties of Beurling's functions H±(z), Selberg observed that the real entire functions 1 n o R±(z) = H±∆(t+z) + H±∆(t−z) (2.1) 2 2 − + satisfy the inequalities R (x) ≤ χ[−t;t](x) ≤ R (x) for all real x and have Fourier transforms supported in [−∆; ∆]. We summarize these (and some other relevant) properties of the functions R±(z) in the following lemma. Lemma 2. Let t and ∆ be positive real numbers. The even real entire functions R±(z) defined in (2.1) satisfy the following properties: − + (i) R (x) ≤ χ[−t;t](x) ≤ R (x) for all real x ; Z 1 ± 1 (ii) R (x) − χ[−t;t](x) dx = ; −∞ ∆ (iii) R±(z) e2π∆jIm zj ; (iv) R±(x) min(1; ∆−2(jxj − t)−2) for jxj > t ; (v) Rb±(ξ) = 0 for jξj ≥ ∆ ; sin 2πtξ 1 (vi) Rb±(ξ) = + O for jξj ≤ ∆. πξ ∆ The implied constants in (iii), (iv) and (vi) are absolute. Proof. This is a specialization of [9, Lemma 2]. For a further discussion on these Beurling-Selberg extremal functions, as well as proofs of the above assertions, see the survey articles of Montgomery [14], Selberg [19] and Vaaler [20]. 3. Proof of Theorem 1 The functional equation for the Riemann zeta-function, in symmetric form, can be written as ΓR(s)ζ(s) = ΓR(1−s)ζ(1−s) s − 2 s where ΓR(s) = π Γ( 2 ). Lemma 3. For all t > 0, we have 1 Z t Γ0 R 1 N(t) = ( 2 +iu) du + S(t) + 1: 2π −t ΓR Proof. Using the functional equation for ζ(s), the argument principle, and the Schwarz reflection principle, it can be shown that 1 N(t) = arg Γ ( 1 +it) + S(t) + 1 π R 2 for any t > 0. See, for instance, [15, Theorem 14.1]. Since Z t Γ0 R 1 1 ( 2 +iu) du = 2 arg ΓR( 2 +it); −t ΓR the lemma follows. 1 Lemma 4. Let h(s) be analytic in the strip jIm sj ≤ 2 + " for some " > 0, and assume that jh(s)j (1 + jsj)−(1+δ) for some δ > 0 when jRe sj ! 1. Then X ρ− 1 1 1 1 Z 1 Γ0 h 2 = h + h − + h(u) Re R ( 1 +iu) du i 2i 2i π Γ 2 ρ −∞ R (3.1) 1 1 X Λ(n) log n log n − p bh + bh − ; 2π n 2π 2π n=2 3 where the sum on the left-hand side runs over the nontrivial zeros ρ of ζ(s) and Λ(n) is the von Mangoldt function defined to be log p if n = pk, p a prime and k ≥ 1, and zero otherwise. Proof. This is [9, Lemma 1]. For a similar formula, see [10, Equation (25.10)]. We now deduce Theorem 1 from Lemmas 2, 3 and 4. Proof of Theorem 1. Assume the Riemann hypothesis and that ∆ ≥ 1 and t ≥ 10. By Lemma 2 (i), we have X n + o X + 0 ≤ R (γ) − χ[−t;t](γ) = R (γ) − 2 N(t) (3.2) γ γ 1 + where the sum runs over the ordinates of the nontrivial zeros ρ = 2 +iγ of ζ(s). Note that R (u) is even and satisfies the conditions of Lemma 4. In fact, the growth condition required in Lemma 4 follows by Lemma 2 (iii)-(iv) via the Phragm´en-Lindel¨ofprinciple. Using inequality (3.2) in conjunction with Lemma 3 and (3.1), we derive that 1 Z 1 n o Γ0 1 1 Λ(n) log n 1 + R 1 X + + S(t) ≤ R (u)−χ[−t;t](u) ( +iu) du − p Rb + R − 1: (3.3) 2π Γ 2 2π n 2π 2i −∞ R n=2 + 1 π∆ Lemma 2 (iii) implies that R 2i e , while Lemma 2 (v)-(vi) imply that the sum over prime powers in (3.3) is X Λ(n) 1 1 1 p + eπ∆ n log n ∆ ∆ n≤e2π∆ by the Prime Number Theorem and partial summation. Therefore 1 Z 1 n o Γ0 + R 1 π∆ S(t) ≤ R (u)−χ[−t;t](u) ( 2 +iu) du + O e : (3.4) 2π −∞ ΓR It remains to estimate the integral in (3.4).
Load more

Recommended publications
  • The Lerch Zeta Function and Related Functions
    The Lerch Zeta Function and Related Functions Je↵ Lagarias, University of Michigan Ann Arbor, MI, USA (September 20, 2013) Conference on Stark’s Conjecture and Related Topics , (UCSD, Sept. 20-22, 2013) (UCSD Number Theory Group, organizers) 1 Credits (Joint project with W. C. Winnie Li) J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function I. Zeta Integrals, Forum Math, 24 (2012), 1–48. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function II. Analytic Continuation, Forum Math, 24 (2012), 49–84. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function III. Polylogarithms and Special Values, preprint. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function IV. Two-variable Hecke operators, in preparation. Work of J. C. Lagarias is partially supported by NSF grants DMS-0801029 and DMS-1101373. 2 Topics Covered Part I. History: Lerch Zeta and Lerch Transcendent • Part II. Basic Properties • Part III. Multi-valued Analytic Continuation • Part IV. Consequences • Part V. Lerch Transcendent • Part VI. Two variable Hecke operators • 3 Part I. Lerch Zeta Function: History The Lerch zeta function is: • e2⇡ina ⇣(s, a, c):= 1 (n + c)s nX=0 The Lerch transcendent is: • zn Φ(s, z, c)= 1 (n + c)s nX=0 Thus ⇣(s, a, c)=Φ(s, e2⇡ia,c). 4 Special Cases-1 Hurwitz zeta function (1882) • 1 ⇣(s, 0,c)=⇣(s, c):= 1 . (n + c)s nX=0 Periodic zeta function (Apostol (1951)) • e2⇡ina e2⇡ia⇣(s, a, 1) = F (a, s):= 1 . ns nX=1 5 Special Cases-2 Fractional Polylogarithm • n 1 z z Φ(s, z, 1) = Lis(z)= ns nX=1 Riemann zeta function • 1 ⇣(s, 0, 1) = ⇣(s)= 1 ns nX=1 6 History-1 Lipschitz (1857) studies general Euler integrals including • the Lerch zeta function Hurwitz (1882) studied Hurwitz zeta function.
    [Show full text]
  • Generalized Riemann Hypothesis Léo Agélas
    Generalized Riemann Hypothesis Léo Agélas To cite this version: Léo Agélas. Generalized Riemann Hypothesis. 2019. hal-00747680v3 HAL Id: hal-00747680 https://hal.archives-ouvertes.fr/hal-00747680v3 Preprint submitted on 29 May 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Generalized Riemann Hypothesis L´eoAg´elas Department of Mathematics, IFP Energies nouvelles, 1-4, avenue de Bois-Pr´eau,F-92852 Rueil-Malmaison, France Abstract (Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most impor- tant unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems (Borwien et al.(2008), Sabbagh(2002)). In this paper, we give a proof of Gen- eralized Riemann Hypothesis which implies the proof of Riemann Hypothesis and Goldbach's weak conjecture (also known as the odd Goldbach conjecture) one of the oldest and best-known unsolved problems in number theory. 1. Introduction The Riemann hypothesis is one of the most important conjectures in math- ematics.
    [Show full text]
  • A Short and Simple Proof of the Riemann's Hypothesis
    A Short and Simple Proof of the Riemann’s Hypothesis Charaf Ech-Chatbi To cite this version: Charaf Ech-Chatbi. A Short and Simple Proof of the Riemann’s Hypothesis. 2021. hal-03091429v10 HAL Id: hal-03091429 https://hal.archives-ouvertes.fr/hal-03091429v10 Preprint submitted on 5 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Short and Simple Proof of the Riemann’s Hypothesis Charaf ECH-CHATBI ∗ Sunday 21 February 2021 Abstract We present a short and simple proof of the Riemann’s Hypothesis (RH) where only undergraduate mathematics is needed. Keywords: Riemann Hypothesis; Zeta function; Prime Numbers; Millennium Problems. MSC2020 Classification: 11Mxx, 11-XX, 26-XX, 30-xx. 1 The Riemann Hypothesis 1.1 The importance of the Riemann Hypothesis The prime number theorem gives us the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper[1], it asserts that all the ’non-trivial’ zeros of the zeta function are complex numbers with real part 1/2. 1.2 Riemann Zeta Function For a complex number s where ℜ(s) > 1, the Zeta function is defined as the sum of the following series: +∞ 1 ζ(s)= (1) ns n=1 X In his 1859 paper[1], Riemann went further and extended the zeta function ζ(s), by analytical continuation, to an absolutely convergent function in the half plane ℜ(s) > 0, minus a simple pole at s = 1: s +∞ {x} ζ(s)= − s dx (2) s − 1 xs+1 Z1 ∗One Raffles Quay, North Tower Level 35.
    [Show full text]
  • The Riemann and Hurwitz Zeta Functions, Apery's Constant and New
    The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving ζ(2k) Cezar Lupu1 1Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra, Combinatorics and Geometry Graduate Student Research Seminar, February 2, 2017, Pittsburgh, PA A quick overview of the Riemann zeta function. The Riemann zeta function is defined by 1 X 1 ζ(s) = ; Re s > 1: ns n=1 Originally, Riemann zeta function was defined for real arguments. Also, Euler found another formula which relates the Riemann zeta function with prime numbrs, namely Y 1 ζ(s) = ; 1 p 1 − ps where p runs through all primes p = 2; 3; 5;:::. A quick overview of the Riemann zeta function. Moreover, Riemann proved that the following ζ(s) satisfies the following integral representation formula: 1 Z 1 us−1 ζ(s) = u du; Re s > 1; Γ(s) 0 e − 1 Z 1 where Γ(s) = ts−1e−t dt, Re s > 0 is the Euler gamma 0 function. Also, another important fact is that one can extend ζ(s) from Re s > 1 to Re s > 0. By an easy computation one has 1 X 1 (1 − 21−s )ζ(s) = (−1)n−1 ; ns n=1 and therefore we have A quick overview of the Riemann function. 1 1 X 1 ζ(s) = (−1)n−1 ; Re s > 0; s 6= 1: 1 − 21−s ns n=1 It is well-known that ζ is analytic and it has an analytic continuation at s = 1. At s = 1 it has a simple pole with residue 1.
    [Show full text]
  • RIEMANN's HYPOTHESIS 1. Gauss There Are 4 Prime Numbers Less
    RIEMANN'S HYPOTHESIS BRIAN CONREY 1. Gauss There are 4 prime numbers less than 10; there are 25 primes less than 100; there are 168 primes less than 1000, and 1229 primes less than 10000. At what rate do the primes thin out? Today we use the notation π(x) to denote the number of primes less than or equal to x; so π(1000) = 168. Carl Friedrich Gauss in an 1849 letter to his former student Encke provided us with the answer to this question. Gauss described his work from 58 years earlier (when he was 15 or 16) where he came to the conclusion that the likelihood of a number n being prime, without knowing anything about it except its size, is 1 : log n Since log 10 = 2:303 ::: the means that about 1 in 16 seven digit numbers are prime and the 100 digit primes are spaced apart by about 230. Gauss came to his conclusion empirically: he kept statistics on how many primes there are in each sequence of 100 numbers all the way up to 3 million or so! He claimed that he could count the primes in a chiliad (a block of 1000) in 15 minutes! Thus we expect that 1 1 1 1 π(N) ≈ + + + ··· + : log 2 log 3 log 4 log N This is within a constant of Z N du li(N) = 0 log u so Gauss' conjecture may be expressed as π(N) = li(N) + E(N) Date: January 27, 2015. 1 2 BRIAN CONREY where E(N) is an error term.
    [Show full text]
  • 2 Values of the Riemann Zeta Function at Integers
    MATerials MATem`atics Volum 2009, treball no. 6, 26 pp. ISSN: 1887-1097 2 Publicaci´oelectr`onicade divulgaci´odel Departament de Matem`atiques MAT de la Universitat Aut`onomade Barcelona www.mat.uab.cat/matmat Values of the Riemann zeta function at integers Roman J. Dwilewicz, J´anMin´aˇc 1 Introduction The Riemann zeta function is one of the most important and fascinating functions in mathematics. It is very natural as it deals with the series of powers of natural numbers: 1 1 1 X 1 X 1 X 1 ; ; ; etc. (1) n2 n3 n4 n=1 n=1 n=1 Originally the function was defined for real argu- ments as Leonhard Euler 1 X 1 ζ(x) = for x > 1: (2) nx n=1 It connects by a continuous parameter all series from (1). In 1734 Leon- hard Euler (1707 - 1783) found something amazing; namely he determined all values ζ(2); ζ(4); ζ(6);::: { a truly remarkable discovery. He also found a beautiful relationship between prime numbers and ζ(x) whose significance for current mathematics cannot be overestimated. It was Bernhard Riemann (1826 - 1866), however, who recognized the importance of viewing ζ(s) as 2 Values of the Riemann zeta function at integers. a function of a complex variable s = x + iy rather than a real variable x. Moreover, in 1859 Riemann gave a formula for a unique (the so-called holo- morphic) extension of the function onto the entire complex plane C except s = 1. However, the formula (2) cannot be applied anymore if the real part of s, Re s = x is ≤ 1.
    [Show full text]
  • Zeta Functions and Chaos Audrey Terras October 12, 2009 Abstract: the Zeta Functions of Riemann, Selberg and Ruelle Are Briefly Introduced Along with Some Others
    Zeta Functions and Chaos Audrey Terras October 12, 2009 Abstract: The zeta functions of Riemann, Selberg and Ruelle are briefly introduced along with some others. The Ihara zeta function of a finite graph is our main topic. We consider two determinant formulas for the Ihara zeta, the Riemann hypothesis, and connections with random matrix theory and quantum chaos. 1 Introduction This paper is an expanded version of lectures given at M.S.R.I. in June of 2008. It provides an introduction to various zeta functions emphasizing zeta functions of a finite graph and connections with random matrix theory and quantum chaos. Section 2. Three Zeta Functions For the number theorist, most zeta functions are multiplicative generating functions for something like primes (or prime ideals). The Riemann zeta is the chief example. There are analogous functions arising in other fields such as Selberg’s zeta function of a Riemann surface, Ihara’s zeta function of a finite connected graph. We will consider the Riemann hypothesis for the Ihara zeta function and its connection with expander graphs. Section 3. Ruelle’s zeta function of a Dynamical System, A Determinant Formula, The Graph Prime Number Theorem. The first topic is the Ruelle zeta function which will be shown to be a generalization of the Ihara zeta. A determinant formula is proved for the Ihara zeta function. Then we prove the graph prime number theorem. Section 4. Edge and Path Zeta Functions and their Determinant Formulas, Connections with Quantum Chaos. We define two more zeta functions associated to a finite graph - the edge and path zetas.
    [Show full text]
  • A RIEMANN HYPOTHESIS for CHARACTERISTIC P L-FUNCTIONS
    A RIEMANN HYPOTHESIS FOR CHARACTERISTIC p L-FUNCTIONS DAVID GOSS Abstract. We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute values. Further we use absolute values to give similar reformulations of the classical conjectures (with, perhaps, finitely many exceptional zeroes). We show how both sets of conjectures behave in remarkably similar ways. 1. Introduction The arithmetic of function fields attempts to create a model of classical arithmetic using Drinfeld modules and related constructions such as shtuka, A-modules, τ-sheaves, etc. Let k be one such function field over a finite field Fr and let be a fixed place of k with completion K = k . It is well known that the algebraic closure1 of K is infinite dimensional over K and that,1 moreover, K may have infinitely many distinct extensions of a bounded degree. Thus function fields are inherently \looser" than number fields where the fact that [C: R] = 2 offers considerable restraint. As such, objects of classical number theory may have many different function field analogs. Classifying the different aspects of function field arithmetic is a lengthy job. One finds for instance that there are two distinct analogs of classical L-series. One analog comes from the L-series of Drinfeld modules etc., and is the one of interest here. The other analog arises from the L-series of modular forms on the Drinfeld rigid spaces, (see, for instance, [Go2]).
    [Show full text]
  • Special Values of Riemann's Zeta Function
    The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 Special values of Riemann's zeta function Cameron Franc UC Santa Cruz March 6, 2013 Cameron Franc Special values of Riemann's zeta function The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 Riemann's zeta function If s > 1 is a real number, then the series X 1 ζ(s) = ns n≥1 converges. Proof: Compare the partial sum to an integral, N X 1 Z N dx 1 1 1 ≤ 1 + = 1 + 1 − ≤ 1 + : ns xs s − 1 Ns−1 s − 1 n=1 1 Cameron Franc Special values of Riemann's zeta function The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 The resulting function ζ(s) is called Riemann's zeta function. Was studied in depth by Euler and others before Riemann. ζ(s) is named after Riemann for two reasons: 1 He was the first to consider allowing the s in ζ(s) to be a complex number 6= 1. 2 His deep 1859 paper \Ueber die Anzahl der Primzahlen unter einer gegebenen Gr¨osse" (\On the number of primes less than a given quantity") made remarkable connections between ζ(s) and prime numbers. Cameron Franc Special values of Riemann's zeta function The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 In this talk we will discuss certain special values of ζ(s) for integer values of s.
    [Show full text]
  • Experimental Observations on the Riemann Hypothesis, and the Collatz Conjecture Chris King May 2009-Feb 2010 Mathematics Department, University of Auckland
    Experimental Observations on the Riemann Hypothesis, and the Collatz Conjecture Chris King May 2009-Feb 2010 Mathematics Department, University of Auckland Abstract: This paper seeks to explore whether the Riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity. It also seeks to provide an experimental basis to discover some of the mathematical enigmas surrounding these conjectures, by providing Matlab and C programs which the reader can use to explore and better understand these systems (see appendix 6). Fig 1: The Riemann functions (z) and (z) : absolute value in red, angle in green. The pole at z = 1 and the non- trivial zeros on x = showing in (z) as a peak and dimples. The trivial zeros are at the angle shifts at even integers on the negative real axis. The corresponding zeros of (z) show in the central foci of angle shift with the absolute value and angle reflecting the function’s symmetry between z and 1 - z. If there is an analytic reason why the zeros are on x = one would expect it to be a manifest property of the reflective symmetry of (z) . Introduction: The Riemann hypothesis1,2 remains the most challenging unsolved problem in mathematics at the beginning of the third millennium. The other two problems of similar status, Fermat’s last theorem3 and the Poincare conjecture4 have both succumbed to solutions by Andrew Wiles and Grigori Perelman in tours de force using a swathe of advanced techniques from diverse mathematical areas. Fermat’s last theorem states that no three integers a, b, c can satisfy an + bn = cn for n > 2.
    [Show full text]
  • Complex Zeros of the Riemann Zeta Function Are on the Critical Line
    December 2020 All Complex Zeros of the Riemann Zeta Function Are on the Critical Line: Two Proofs of the Riemann Hypothesis (°) Roberto Violi (*) Dedicated to the Riemann Hypothesis on the Occasion of its 160th Birthday (°) Keywords: Riemann Hypothesis; Riemann Zeta-Function; Dirichlet Eta-Function; Critical Strip; Zeta Zeros; Prime Number Theorem; Millennium Problems; Dirichlet Series; Generalized Riemann Hypothesis. Mathematics Subject Classification (2010): 11M26, 11M41. (*) Banca d’Italia, Financial Markets and Payment System Department, Rome, via Nazionale 91, Italy. E-Mail: [email protected]. The opinions of this article are those of the author and do not reflect in any way the views or policies of the Banca d’Italia or the Eurosystem of Central Banks. All errors are mine. Abstract In this study I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta-Function is the critical line given by 1 ℜ(s) = . The methods and results of this paper are based on well-known theorems 2 on the number of zeros for complex-valued functions (Jensen’s, Titchmarsh’s and Rouché’s theorem), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well- known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta-Function, except leveraging its implied symmetry for non-trivial zeros on the critical strip.
    [Show full text]
  • The Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
    Math 259: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler's identity: 1 1 1 s 0 cps1 [ζ(s) :=] n− = p− = s : (1) X Y X Y 1 p− n=1 p prime @cp=1 A p prime − We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann's insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it: s σ We already observed that for all real n > 0 we have n− = n− , because j j s σ it log n n− = exp( s log n) = n− e − and eit log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the \Riemann zeta function"), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula n+1 n+1 1 1 s s 1 s s ζ(s) = n− Z x− dx = Z (n− x− ) dx; − s 1 X − X − − n=1 n n=1 n since for x [n; n + 1] (n 1) and σ > 0 we have 2 ≥ x s s 1 s 1 σ n− x− = s Z y− − dy s n− − j − j ≤ j j n so the formula for ζ(s) (1=(s 1)) is a sum of analytic functions converging absolutely in compact subsets− of− σ + it : σ > 0 and thus gives an analytic function there.
    [Show full text]