DOCSLIB.ORG

A Short and Simple Proof of the Riemann's Hypothesis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. 048583 Singapore. Email: [email protected]. The opinions of this article are those of the author and do not reflect in any way the views or business of his employer. 1 Where {x} = x − [x] is the fractional part and [x] is the integer part of x. There is another way [2] to analytically continue ζ(s) to the region ℜ(s) > 0. The idea is to observe that for ℜ(s) > 1: +∞ +∞ 2 1 2 (1 − )ζ(s) = − (3) 2s ns (2n)s n=1 n=1 X X Thus, +∞ 2 (−1)n+1 ζ(s) = (1 − )−1 (4) 2s ns n=1 X 2 = (1 − )−1η(s) (5) 2s The Dirichlet eta function η(s) converges conditionally when ℜ(s) > 0 2kπ and s 6= 1+ i ln(2) . η(s) is used as analytical continuation of the Zeta function on the domain where ℜ(s) > 0. Riemann also obtained the analytic continuation of the zeta function to the whole complex plane. Riemann[1] has shown that Zeta has a functional equation1 s s πs ζ(s) = 2 π −1 sin Γ(1 − s)ζ(1 − s) (7) 2 Where Γ(s) is the Gamma function. Using the above functional equa- tion, Riemann has shown that the non-trivial zeros of ζ are located sym- metrically with respect to the line ℜ(s) = 1/2, inside the critical strip 0 < ℜ(s) < 1. Riemann has conjectured that all the non trivial-zeros are located on the critical line ℜ(s) = 1/2. In 1921, Hardy & Littlewood[2,3, 6] showed that there are infinitely many zeros on the critical line. In 1896, Hadamard and De la Vall´ee Poussin[2,3] independently proved that ζ(s) has no zeros of the form s =1+ it for t ∈ R. Some of the known results[2, 3] of ζ(s) are as follows: • ζ(s) has no zero for ℜ(s) > 1. • ζ(s) has no zero of the form s =1+ iτ. i.e. ζ(1 + iτ) 6= 0, ∀ τ. • ζ(s) has a simple pole at s = 1 with residue 1. • ζ(s) has all the trivial zeros at the negative even integers s = −2k, k ∈ N∗. • The non-trivial zeros are inside the critical strip: i.e. 0 < ℜ(s) < 1. • If ζ(s) = 0, then 1 − s,s ¯ and 1 − s¯ are also zeros of ζ: i.e. ζ(s) = ζ(1 − s)= ζ(¯s)= ζ(1 − s¯) = 0. Therefore, to prove the “Riemann Hypothesis” (RH), it is sufficient to prove that ζ has no zero on the right hand side 1/2 < ℜ(s) < 1 of the critical strip. 1This is slightly different from the functional equation presented in Riemann’s paper[1]. This is a variation that is found everywhere in the litterature[2,3,4]. Another variant using the cos: s s πs ζ(1 − s) = 21− π− cos Γ(s)ζ(s) (6) 2 2 1.3 Proof of the Riemann Hypothesis Let’s take a complex number s such that s = σ + iτ. Unless we explicitly mention otherwise, let’s suppose that 0 <σ< 1, τ > 0 and ζ(s) = 0. We have from the Riemann’s integral above: s +∞ {x} ζ(s)= − s dx (8) s − 1 xs+1 Z1 We have s 6= 1, s 6= 0 and ζ(s) = 0, therefore: 1 +∞ {x} = dx (9) s − 1 xs+1 Z1 Therefore: 1 +∞ {x} = dx (10) σ + iτ − 1 xσ+iτ+1 Z1 And σ − 1 − iτ +∞ cos(τ ln (x)) − i sin(τ ln (x)) {x} = dx (11) (σ − 1)2 + τ 2 xσ+1 Z1 The integral is absolutely convergent. We take the real part and the imaginary part of both sides of the above equation and define the functions F and G as following: σ − 1 F (σ, τ) = (12) (σ − 1)2 + τ 2 +∞ cos(τ ln (x)) {x} = dx (13) xσ+1 Z1 And τ G(σ, τ) = (14) (σ − 1)2 + τ 2 +∞ sin(τ ln (x)) {x} = dx (15) xσ+1 Z1 We also have 1 − s¯ = 1 − σ + iτ = σ1 + iτ1 a zero for ζ with a real part σ1 such that 0 <σ1 = 1 − σ < 1 and an imaginary part τ1 such that τ1 = τ. Therefore σ1 − 1 F (1 − σ, τ) = 2 2 (16) (σ1 − 1) + τ1 −σ = (17) σ2 + τ 2 +∞ cos(τ ln (x)) {x} = dx (18) x2−σ Z1 3 And τ1 G(1 − σ, τ) = 2 2 (19) (σ1 − 1) + τ1 τ = (20) σ2 + τ 2 +∞ sin(τ ln (x)) {x} = dx (21) x2−σ Z1 Before we move forward, we need to define the following function I(a,σ,τ) for a> 0: a sin(τ ln(x)) I(a,σ,τ) = dx (22) xσ Z1 ln(a) σ x = sin(τx)e(1− ) dx (23) Z0 cos(τ ln(a)) (σ − 1) sin(τ ln(a)) = K(σ, τ) 1 − − (24) aσ−1 τ aσ−1 Where τ K(σ, τ) = (25) (σ − 1)2 + τ 2 Let’s define the function J(a,σ,τ) for a> 0: a cos(τ ln(x)) J(a,σ,τ) = dx (26) xσ Z1 ln(a) σ x = cos(τx)e(1− ) dx (27) Z0 (σ − 1) (σ − 1) cos(τ ln(a)) sin(τ ln(a)) = K(σ, τ) − + (28) τ τ aσ−1 aσ−1 Where K(σ, τ) is defined above in the equation (25). Now, let’s write G(σ, τ) as the limit of a sequence as follows: N {x} sin(τ ln(x)) G(σ, τ) = lim dx (29) N 1+σ →+∞ 1 x Z τ N x σ x ln( ) ǫ(e τ )e− τ = lim dx sin(x) (30) N→+∞ τ Z0 τ ln(N) = lim dxg(x) sin(x) (31) N→+∞ Z0 = lim U(N,σ) (32) N→+∞ Where τ ln(N) U(N,σ) = dxg(x) sin(x) (33) 0 Z x ǫ(e τ ) σ x g(x) = e− τ (34) τ ǫ(x) = {x} (35) 4 And the same for F (σ, τ): N {x} cos(τ ln(x)) F (σ, τ) = lim dx (36) N 1+σ →+∞ 1 x Z τ N x σ x ln( ) ǫ(e τ )e− τ = lim dx cos(x) (37) N→+∞ τ Z0 τ ln(N) = lim dxg(x) cos(x) (38) N→+∞ Z0 = lim V (N,σ) (39) N→+∞ Where τ ln(N) V (N,σ) = dxg(x) cos(x) (40) Z0 Let’s study the function g over R. We have the function g piecewise continuous and and its derivatives are also peicewise continuous and: x ǫ(e τ ) σ x g(x) = e− τ (41) τ 1−σ x e τ x σ g′(x) = ǫ′(e τ ) − g(x) (42) τ 2 τ The interval (0,τ ln(N)) is the union of the intervals (τ ln(n),τ ln(n + 1)) where 1 ≤ n < N. We have ǫ′(x) = 1 for each x ∈ (τ log(n),τ log(n + 1)) for each 1 ≤ n < N. Therefore, thanks to the integration by parts we can write: τ ln(N) U(N,σ) = dxg(x) sin(x) (43) Z0 N−1 = g(0+) cos(0) + lim g(τ ln(n)+ ǫ) cos(τ ln(n)+ ǫ) − g(τ ln(n) − ǫ) cos(τ ln(n) − ǫ) (44) ǫ→0 n=1 X i τ ln(N) −g(τ ln(N)−) cos(τ ln(N)) + dxg′(x) cos(x) (45) Z0 N−1 = lim g(τ ln(n)+ ǫ) − g(τ ln(n) − ǫ) cos(τ ln(n)) (46) ǫ→0 n =1 i X τ N cos(τ ln(N)) ln( ) − + dxg′(x) cos(x) (47) τN σ Z0 N−1 τ N 1 cos(τ ln(n)) cos(τ ln(N)) ln( ) = − − + dxg′(x) cos(x) (48) τ nσ τN σ n=1 0 X Z N τ N τ N 1−σ x 1 cos(τ ln(n)) σ ln( ) ln( ) e τ x = − − dxg(x) cos(x)+ dx ǫ′(e τ ) cos(x) (49) τ nσ τ τ 2 n=1 0 0 ! X Z Z N τ N 1−σ x 1 cos(τ ln(n)) σ ln( ) e τ = − − V (N,σ)+ dx cos(x) (50) τ nσ τ τ 2 n=1 0 X Z 5 Therefore N σ 1 cos(τ ln(n)) 1 U(N,σ)+ V (N,σ) = − + J(N,σ,τ) (51) τ τ nσ τ n=1 X Therefore N cos(τ ln(n)) − J(N,σ,τ) = −τU(N,σ) − σV (N,σ) (52) nσ n=1 X N cos(τ ln(n)) Therefore, the sequence n=1 nσ −J(N,σ,τ) converges N≥1 and its limit is as follows: P N cos(τ ln(n)) τ 2 + σ(σ − 1) lim σ − J(N,σ,τ) = − (53) N→+∞ n (1 − σ)2 + τ 2 n=1 X Therefore we can write the following: N cos(τ ln(n)) τ 2 + σ(σ − 1) = J(N,σ,τ) − + αN (54) nσ (1 − σ)2 + τ 2 n=1 X Where (αN ) is a bounded sequence with limit zero.
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]
  • An Amazing Prime Heuristic.Pdf
    This document has been moved to https://arxiv.org/abs/2103.04483 Please use that version instead. AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 264955 1: · The very next day Giovanni La Barbera found the new record primes: 1693965 266443 1: · The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to N dx 2C2N 2C2 2 2 Z2 (log x) ∼ (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture.
    [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]
  • Bayesian Perspectives on Mathematical Practice
    Bayesian perspectives on mathematical practice James Franklin University of New South Wales, Sydney, Australia In: B. Sriraman, ed, Handbook of the History and Philosophy of Mathematical Practice, Springer, 2020. Contents 1. Introduction 2. The relation of conjectures to proof 3. Applied mathematics and statistics: understanding the behavior of complex models 4. The objective Bayesian perspective on evidence 5. Evidence for and against the Riemann Hypothesis 6. Probabilistic relations between necessary truths? 7. The problem of induction in mathematics 8. Conclusion References Abstract Mathematicians often speak of conjectures as being confirmed by evi- dence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long re- sisted proof, such as the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in com- puter power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore be- come (among other things) an experimental science (though that has not dimin- ished the importance of proof in the traditional style). We examine how the evalu- ation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Nu- merical evidence in mathematics is related to the problem of induction; the occur- rence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical rea- soning.
    [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]