A Appendix: a Short History of Universality
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Dirichlet Series
数理解析研究所講究録 1088 巻 1999 年 83-93 83 Overconvergence Phenomena For Generalized Dirichlet Series Daniele C. Struppa Department of Mathematical Sciences George Mason University Fairfax, VA 22030 $\mathrm{U}.\mathrm{S}$ . A 84 1 Introduction This paper contains the text of the talk I delivered at the symposium “Resurgent Functions and Convolution Equations”, and is an expanded version of a joint paper with T. Kawai, which will appear shortly and which will contain the full proofs of the results. The goal of our work has been to provide a new approach to the classical topic of overconvergence for Dirichlet series, by employing results in the theory of infinite order differential operators with constant coefficients. The possibility of linking infinite order differential operators with gap theorems and related subjects such as overconvergence phenomena was first suggested by Ehrenpreis in [6], but in a form which could not be brought to fruition. In this paper we show how a wide class of overconvergence phenomena can be described in terms of infinite order differential operators, and that we can provide a multi-dimensional analog for such phenomena. Let us begin by stating the problem, as it was first observed by Jentzsch, and subsequently made famous by Ostrowski (see [5]). Consider a power series $f(z)= \sum_{=n0}^{+\infty}anZ^{n}$ (1) whose circle of convergence is $\triangle(0, \rho)=\{z\in C : |z|<\rho\}$ , with $\rho$ given, therefore, by $\rho=\varliminf|a_{n}|^{-1/}n$ . Even though we know that the series given in (1) cannot converge outside of $\Delta=\Delta(0, \rho)$ , it is nevertheless possible that some subsequence of its sequence of partial sums may converge in a region overlapping with $\triangle$ . -
Wei Zhang – Positivity of L-Functions and “Completion of the Square”
Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Positivity of L-functions and “Completion of square" Wei Zhang Massachusetts Institute of Technology Bristol, June 4th, 2018 1 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Outline 1 Riemann hypothesis 2 Positivity of L-functions 3 Completion of square 4 Positivity on surfaces 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Riemann Hypothesis (RH) Riemann zeta function 1 X 1 1 1 ζ(s) = = 1 + + + ··· ns 2s 3s n=1 Y 1 = ; s 2 ; Re(s) > 1 1 − p−s C p; primes Analytic continuation and Functional equation −s=2 Λ(s) := π Γ(s=2)ζ(s) = Λ(1 − s); s 2 C Conjecture The non-trivial zeros of the Riemann zeta function ζ(s) lie on the line 1 Re(s) = : 3 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Riemann Hypothesis (RH) Riemann zeta function 1 X 1 1 1 ζ(s) = = 1 + + + ··· ns 2s 3s n=1 Y 1 = ; s 2 ; Re(s) > 1 1 − p−s C p; primes Analytic continuation and Functional equation −s=2 Λ(s) := π Γ(s=2)ζ(s) = Λ(1 − s); s 2 C Conjecture The non-trivial zeros of the Riemann zeta function ζ(s) lie on the line 1 Re(s) = : 3 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Equivalent statements of RH Let π(X) = #fprimes numbers p ≤ Xg: Then Z X dt 1=2+ RH () π(X) − = O(X ): 2 log t Let X θ(X) = log p: p<X Then RH () jθ(X) − Xj = O(X 1=2+): 4 Riemann hypothesis Positivity of L-functions Completion of square -
Lagrange Inversion Theorem for Dirichlet Series Arxiv:1911.11133V1
Lagrange Inversion Theorem for Dirichlet series Alexey Kuznetsov ∗ November 26, 2019 Abstract We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4]. Keywords: Dirichlet series, Lagrange Inversion Theorem, convolution polynomials 2010 Mathematics Subject Classification : Primary 30B50, Secondary 11M41 1 Introduction 0 The Lagrange Inversion Theorem states that if φ(z) is analytic near z = z0 with φ (z0) 6= 0 then the function η(w), defined as the solution to φ(η(w)) = w, is analytic near w0 = φ(z0) and is represented there by the power series X bn η(w) = z + (w − w )n; (1) 0 n! 0 n≥0 where the coefficients are given by n−1 n d z − z0 bn = lim n−1 : (2) z!z0 dz φ(z) − φ(z0) 0 This result can be stated in the following simpler form. We assume that φ(z0) = z0 = 0 and φ (0) = 1. We define A(z) = z=φ(z) and check that A is analytic near z = 0 and is represented there by the power series arXiv:1911.11133v1 [math.NT] 25 Nov 2019 X n A(z) = 1 + anz : (3) n≥1 Let us define B(z) implicitly via A(zB(z)) = B(z): (4) Then the function η(z) = zB(z) solves equation η(z)=A(η(z)) = z, which is equivalent to φ(η(z)) = z. Applying Lagrange Inversion Theorem in the form (2) we obtain X an(n + 1) B(z) = zn; (5) n + 1 n≥0 ∗Dept. -
Notes on the Riemann Hypothesis Ricardo Pérez-Marco
Notes on the Riemann Hypothesis Ricardo Pérez-Marco To cite this version: Ricardo Pérez-Marco. Notes on the Riemann Hypothesis. 2018. hal-01713875 HAL Id: hal-01713875 https://hal.archives-ouvertes.fr/hal-01713875 Preprint submitted on 21 Feb 2018 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. NOTES ON THE RIEMANN HYPOTHESIS RICARDO PEREZ-MARCO´ Abstract. Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture. Contents 1. Euler transalgebraic world. 2 2. Riemann's article. 8 2.1. Meromorphic extension. 8 2.2. Value at negative integers. 10 2.3. First proof of the functional equation. 11 2.4. Second proof of the functional equation. 12 2.5. The Riemann Hypothesis. 13 2.6. The Law of Prime Numbers. 17 3. On Riemann zeta-function after Riemann. -
Of Triangles, Gas, Price, and Men
OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard -
Interplay Between Number Theory and Analysis for Dirichlet Series
Mathematisches Forschungsinstitut Oberwolfach Report No. 51/2017 DOI: 10.4171/OWR/2017/51 Mini-Workshop: Interplay between Number Theory and Analysis for Dirichlet Series Organised by Fr´ed´eric Bayart, Aubi`ere Kaisa Matom¨aki, Turku Eero Saksman, Helsinki Kristian Seip, Trondheim 29 October – 4 November 2017 Abstract. In recent years a number of challenging research problems have crystallized in the analytic theory of Dirichlet series and its interaction with function theory in polydiscs. Their solutions appear to require unconventional combinations of expertise from harmonic, functional, and complex analysis, and especially from analytic number theory. This MFO workshop provided an ideal arena for the exchange of ideas needed to nurture further progress and to solve important problems. Mathematics Subject Classification (2010): 11xx, 30xx. Introduction by the Organisers The workshop ’Interplay between number theory and analysis for Dirichlet series’, organised by Fr´ed´eric Bayart (Clermont Universit´e), Kaisa Matom¨aki (Univer- sity of Turku), Eero Saksman (University of Helsinki) and Kristian Seip (NTNU, Trondheim) was held October 29th – November 4th, 2017. This meeting was well attended with around 25 participants coming from a number of different countries, including participants form North America. The group formed a nice blend of re- searchers with somewhat different mathematical backgrounds which resulted in a fruitful interaction. About 17 talks, of varying lengths, were delivered during the five days. The talks were given by both leading experts in the field as well as by rising young re- searchers. Given lectures dealt with e.g. connections between number theory and 3036 Oberwolfach Report 51/2017 random matrix theory, operators acting on Dirichlet series, distribution of Beurl- ing primes, growth of Lp-norms of Dirichlet polynomials, growth and density of values of the Riemann zeta on boundary of the critical strip, Rado’s criterion for kth powers, Sarnak’s and Elliot’s conjectures, and Hardy type spaces of general Dirichlet series. -
Analytic Continuation of Ζ(S) Violates the Law of Non-Contradiction (LNC)
Analytic Continuation of ζ(s) Violates the Law of Non-Contradiction (LNC) Ayal Sharon ∗y July 30, 2019 Abstract The Dirichlet series of ζ(s) was long ago proven to be divergent throughout half-plane Re(s) ≤ 1. If also Riemann’s proposition is true, that there exists an "expression" of ζ(s) that is convergent at all s (except at s = 1), then ζ(s) is both divergent and convergent throughout half-plane Re(s) ≤ 1 (except at s = 1). This result violates all three of Aristotle’s "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non- Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion" / Ex Contradictione Quodlibet (ECQ). In addition, the Hankel contour used in Riemann’s analytic continuation of ζ(s) violates Cauchy’s integral theorem, providing another proof of the invalidity of Riemann’s ζ(s). Riemann’s ζ(s) is one of the L-functions, which are all in- valid due to analytic continuation. This result renders unsound all theorems (e.g. Modularity, Fermat’s last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an L-function (e.g. Riemann’s ζ(s)) is valid. We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar’s 3VL, Priest’s LP ). ∗Patent Examiner, U.S. Patent and Trademark Office (USPTO). -
A Connection Between the Riemann Hypothesis and Uniqueness
A Connection between the Riemann Hypothesis and Uniqueness of the Riemann zeta function Pei-Chu Hu and Bao Qin Li Department of Mathematics, Shandong University, Jinan 250100, Shandong, P. R. China E-mail: [email protected] Department of Mathematics and Statistics, Florida International University, Miami, FL 33199 USA E-mail: libaoqin@fiu.edu Abstract In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L- functions. 1 Introduction The Riemann ζ function is defined by the Dirichlet series ∞ arXiv:1610.01583v1 [math.NT] 5 Oct 2016 1 ζ(s)= , s = σ + it (1.1) ns nX=1 Mathematics Subject Classification 2000 (MSC2000): 11M36, 30D35. Key words and phrases: Riemann zeta function; Riemann hypothesis, uniqueness, the Dedekind zeta function, L-function, Riemann functional equation. 1 for Re(s) > 1, which is absolutely convergent, and admits an analytical continuation as a meromorphic function in the complex plane C of order 1, which has only a simple pole at s = 1 with residue equal to 1. It satisfies the following Riemann functional equation: πs ζ(1 s) = 2(2π)−s cos Γ(s)ζ(s), (1.2) − 2 where Γ is the Euler gamma function ∞ Γ(z)= tz−1e−tdt, Rez > 0, Z0 analytically continued as a meromorphic function in C of order 1 without any zeros and with simple poles at s = 0, 1, 2, . − − · · · The allied function s s s ξ(s)= (s 1)π− 2 Γ ζ(s) (1.3) 2 − 2 is an entire function of order equal to 1 satisfying the functional equation ξ(1 s)= ξ(s) (1.4) − (see e.g. -
Fundamental Domains and Analytic Continuation of General Dirichlet Series
FUNDAMENTAL DOMAINS AND ANALYTIC CONTINUATION OF GENERAL DIRICHLET SERIES Dorin Ghisa1 with an introduction by Les Ferry2 AMS subject classification: 30C35, 11M26 Abstract: Fundamental domains are found for functions defined by general Dirichlet series and by using basic properties of conformal mappings the Great Riemann Hypothesis is studied. Introduction My interest in the topic of this paper appeared when I became aware that John Derbyshare, with whom I was in the same class as an undergraduate, had written a book on the Riemann Hypothesis and that this had won plaudits for its remarkable success in representing an abstruse topic to a lay audience. On discovering this area of mathematics for myself through different other sources, I found it utterly fascinating. More than a quarter of century before, I was taken a strong interest in the work of Peitgen, Richter, Douady and Hubbard concerning Julia, Fatou and Mandelbrot sets. From this I had learned the basic truth that, however obstruse might be the method of definition, the nature of analytic function is very tightly controlled in practice. I sensed intuitively at that time the Riemann Zeta function would have to be viewed in this light if a proof of Riemann Hypothesis was to be achieved. Many mathematicians appear to require that any fundamental property of the Riemann Zeta function and in general of any L-function must be derived from the consideration of the Euler product and implicitly from some mysterious properties of prime numbers. My own view is that it is in fact the functional equation which expresses the key property with which any method of proof of the Riemann Hypothesis and its extensions must interact. -
In Presenting the Dissertation As a Partial Fulfillment of The
In presenting the dissertation as a partial fulfillment of the requirements for an advanced degree from the Georgia Institute of Technology, I agree that the Library of the Institute shall make it available for inspection and circulation in accordance with its regulations governing materials of this type. I agree that permission to copy from, or to publish from, this dissertation may be granted by the professor under whose direction it was written, or, in his absence, by the Dean of the Graduate Division when such copying or publication is solely for scholarly purposes and does not involve potential financial gain. It is under stood that any copying from, or publication of, this dis sertation which involves potential financial gain will not be allowed without written permission. .1 I 7/25/6Q DIRICHLET SERIES AS A GENERALIZATION OF POWER SERIES A THESIS Presented to The Faculty of the Graduate Division by Hugh Alexander Sanders In Partial Fulfillment of the Requirements for the Degree Master of Science in Applied Mathematics Georgia Institute of Technology June, 1969 DIRICHLET SERIES AS A GENERALIZATION OF POWER SERIES Approved: Chairman rs O y . "All generalizations are dangerous, even this one. —Alexandre Dumas iii ACKNOWLEDGMENTS I would like to express my grateful appreciation to Dr. James M. Osborn, my thesis advisor, for his hard work, good humor, and expert guidance in helping me complete this thesis. I wish to thank the other members of my reading committee, Professor John P. Line and Dr. James T. Wang, for their many helpful suggestions. I also want to thank Mrs. -
The Riemann Hypothesis the American Institute of Mathematics
The Riemann Hypothesis The American Institute of Mathematics This is a hard{copy version of a web page available through http://www.aimath.org Input on this material is welcomed and can be sent to [email protected] Version: Thu Jun 17 05:56:54 2004 This document is a preliminary version of a planned comprehensive resource on the Riemann Hypothesis. Suggestions and contributions are welcome and can be sent to rh(at)aimath.org 1 2 Table of Contents A. What is an L-function? . 4 1. Terminology and basic properties a. Functional equation b. Euler product c. » and Z functions d. Critical line and critical strip e. Trivial zeros f. Zero counting functions 2. Arithmetic L-functions a. The Riemann zeta function b. Dirichlet L-functions c. Dedekind zeta functions d. GL(2) L-functions e. Higher rank L-functions 3. The Selberg class a. Dirichlet series b. Analytic Continuation c. Functional Equation d. Euler Product e. Ramanujan Hypothesis f. Selberg Conjectures 4. Analogues of zeta-functions a. Dynamical zeta-functions b. Spectral zeta functions B. Riemann Hypotheses . 14 1. Riemann Hypotheses for global L-functions a. The Riemann Hypothesis b. The Generalized Riemann Hypothesis c. The Extended Riemann Hypothesis d. The Grand Riemann Hypothesis 2. Other statements about the zeros of L-functions a. Quasi Riemann Hypothesis b. 100 percent hypothesis c. The Density Hypothesis d. Zeros on the σ = 1 line e. Landau-Siegel zeros f. The vertical distribution of zeros 3. The Lindelof hypothesis and breaking convexity 4. Perspectives on RH a. Analytic number theory b. -
Arxiv:1907.12323V1 [Math.FA]
HENRY HELSON MEETS OTHER BIG SHOTS – A BRIEF SURVEY ANDREAS DEFANT AND INGO SCHOOLMANN Abstract. A theorem of Henry Helson shows that for every ordinary Dirichlet −s series ann with a square summable sequence (an) of coefficients, almost −s all vertical limits anχ(n)n , where χ : N → T is a completely multiplicative arithmeticP function, converge on the right half-plane. We survey on recent improvements andP extensions of this result within Hardy spaces of Dirichlet series – relating it with some classical work of Bohr, Banach, Carleson-Hunt, Ces`aro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann. 1. Introduction In his article [20] from 1967 Henry Helson suggested that the classical theory of ordinary Dirichlet series should be combined with modern harmonic analysis and functional analysis, and since then this theory saw a remarkable comeback – in particular after the publication of the seminal articel [18]. The aim of this survey is to discuss various recent variants of a somewhat −s curious theorem of Helson dealing with Dirichlet series D = ann which have 2-summable coefficients. These Dirichlet series form the natural Hilbert space H2, and to see a very first example look for ε > 0 at the followingP translated zeta series 1 −s (1) D = 1 n . n 2 +ε X In general, each D ∈ H2 converges for all s ∈ C in the half-plane [Re > 1/2]. But if we multiply the an’s with some character χ, i.e. a completely multiplicative arithmetic functions χ : N → T, and consider the new Dirichlet series Dχ = arXiv:1907.12323v1 [math.FA] 29 Jul 2019 −s anχ(n)n , then the convergence in general improves considerably.