Eigenvalues of the Redheffer Matrix and Their Relation to the Mertens Function
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Language Reference Manual
MATLAB The Language5 of Technical Computing Computation Visualization Programming Language Reference Manual Version 5 How to Contact The MathWorks: ☎ (508) 647-7000 Phone PHONE (508) 647-7001 Fax FAX ☎ (508) 647-7022 Technical Support Faxback Server FAX SERVER ✉ MAIL The MathWorks, Inc. Mail 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com Web INTERNET ftp.mathworks.com Anonymous FTP server @ [email protected] Technical support E-MAIL [email protected] Product enhancement suggestions [email protected] Bug reports [email protected] Documentation error reports [email protected] Subscribing user registration [email protected] Order status, license renewals, passcodes [email protected] Sales, pricing, and general information Language Reference Manual (November 1996) COPYRIGHT 1994 - 1996 by The MathWorks, Inc. All Rights Reserved. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro- duced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the software on behalf of any unit or agency of the U. S. Government, the following shall apply: (a) for units of the Department of Defense: RESTRICTED RIGHTS LEGEND: Use, duplication, or disclosure by the Government is subject to restric- tions as set forth in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software Clause at DFARS 252.227-7013. (b) for any other unit or agency: NOTICE - Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction and disclosure are as set forth in Clause 52.227-19(c)(2) of the FAR. -
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. -
Two Elementary Formulae and Some Complicated Properties for Mertens Function3
TWO ELEMENTARY FORMULAE AND SOME COMPLICATED PROPERTIES FOR MERTENS FUNCTION RONG QIANG WEI Abstract. Two elementary formulae for Mertens function M(n) are obtained. With these formulae, M(n) can be calculated directly and simply, which can be easily imple- mented by computer. M(1) ∼ M(2 × 107) are calculated one by one. Based on these 2 × 107 samples, some of the complicated properties for Mertens function M(n), its 16479 zeros, the 10043 local maximum/minimum between two neighbor zeros, and the rela- tion with the cumulative sum of the squarefree integers, are understood numerically and empirically. Keywords: Mertens function Elementary formula Zeros Local maximum/minimum squarefree integers 1. Introduction The Mertens function M(n) is important in number theory, especially the calculation of this function when its argument is large. M(n) is defined as the cumulative sum of the M¨obius function µ(k) for all positive integers n, n (1) M(n)= µ(k) Xk=1 where the M¨obius function µ(k) is defined for a positive integer k by 1 k = 1 (2) µ(k)= 0 if k is divisible by a prime square ( 1)m if k is the product of m distinct primes − Sometimes the above definition can be extended to real numbers as follows: arXiv:1612.01394v2 [math.NT] 15 Dec 2016 (3) M(x)= µ(k) 1≤Xk≤x Furthermore, Mertens function has other representations. They are mainly shown in formula (4-7): 1 c+i∞ xs (4) M(x)= ds 2πi Zc−i∞ sζ(s) 1 2 RONG QIANG WEI where ζ(s) is the the Riemann zeta function, s is complex, and c> 1 (5) M(n)= exp(2πia) aX∈Fn where Fn is the Farey sequence of order n. -
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. -
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. -
Kung J., Rota G.C., Yan C.H. Combinatorics.. the Rota Way
This page intentionally left blank P1: ... CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Combinatorics: The Rota Way Gian-Carlo Rota was one of the most original and colorful mathematicians of the twentieth century. His work on the foundations of combinatorics focused on revealing the algebraic structures that lie behind diverse combinatorial areas and created a new area of algebraic combinatorics. His graduate courses influenced generations of students. Written by two of his former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially or- dered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Mobius¨ functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exer- cises and research problems are included and unexplored areas of possible research are discussed. This book should be on the shelf of all students and researchers in combinatorics and related areas. joseph p. s. kung is a professor of mathematics at the University of North Texas. He is currently an editor-in-chief of Advances in Applied Mathematics. gian-carlo rota (1932–1999) was a professor of applied mathematics and natural philosophy at the Massachusetts Institute of Technology. He was a member of the Na- tional Academy of Science. He was awarded the 1988 Steele Prize of the American Math- ematical Society for his 1964 paper “On the Foundations of Combinatorial Theory I. -
The Generalized Dedekind Determinant
Contemporary Mathematics Volume 655, 2015 http://dx.doi.org/10.1090/conm/655/13232 The Generalized Dedekind Determinant M. Ram Murty and Kaneenika Sinha Abstract. The aim of this note is to calculate the determinants of certain matrices which arise in three different settings, namely from characters on finite abelian groups, zeta functions on lattices and Fourier coefficients of normalized Hecke eigenforms. Seemingly disparate, these results arise from a common framework suggested by elementary linear algebra. 1. Introduction The purpose of this note is three-fold. We prove three seemingly disparate results about matrices which arise in three different settings, namely from charac- ters on finite abelian groups, zeta functions on lattices and Fourier coefficients of normalized Hecke eigenforms. In this section, we state these theorems. In Section 2, we state a lemma from elementary linear algebra, which lies at the heart of our three theorems. A detailed discussion and proofs of the theorems appear in Sections 3, 4 and 5. In what follows below, for any n × n matrix A and for 1 ≤ i, j ≤ n, Ai,j or (A)i,j will denote the (i, j)-th entry of A. A diagonal matrix with diagonal entries y1,y2, ...yn will be denoted as diag (y1,y2, ...yn). Theorem 1.1. Let G = {x1,x2,...xn} be a finite abelian group and let f : G → C be a complex-valued function on G. Let F be an n × n matrix defined by F −1 i,j = f(xi xj). For a character χ on G, (that is, a homomorphism of G into the multiplicative group of the field C of complex numbers), we define Sχ := f(s)χ(s). -
Bounds of the Mertens Functions
Advances in Dynamical Systems and Applications. ISSN 0973-5321, Volume 16, Number 1, (2021) pp. 35-44 © Research India Publications https://dx.doi.org/10.37622/ADSA/16.1.2021.35-44 Bounds of the Mertens Functions Darrell Coxa, Sourangshu Ghoshb, Eldar Sultanowc aGrayson County College, Denison, TX 75020, USA bDepartment of Civil Engineering , Indian Institute of Technology Kharagpur, West Bengal, India cPotsdam University, 14482 Potsdam, Germany Abstract In this paper we derive new properties of Mertens function and discuss about a likely upper bound of the absolute value of the Mertens function √log(푥!) > |푀(푥)| when 푥 > 1. Using this likely bound we show that we have a sufficient condition to prove the Riemann Hypothesis. 1. INTRODUCTION We define the Mobius Function 휇(푘). Depending on the factorization of n into prime factors the function can take various values in {−1, 0, 1} 휇(푛) = 1 if n has even number of prime factors and it is also square- free(divisible by no perfect square other than 1) 휇(푛) = −1 if n has odd number of prime factors and it is also square-free 휇(푛) = 0 if n is divisible by a perfect square. 푛 Mertens function is defined as 푀(푛) = ∑푘=1 휇(푘) where 휇(푘) is the Mobius function. It can be restated as the difference in the number of square-free integers up to 푥 that have even number of prime factors and the number of square-free integers up to 푥 that have odd number of prime factors. The Mertens function rather grows very slowly since the Mobius function takes only the value 0, ±1 in both the positive and 36 Darrell Cox, Sourangshu Ghosh, Eldar Sultanow negative directions and keeps oscillating in a chaotic manner. -
A Matrix Inequality for Möbius Functions
Volume 10 (2009), Issue 3, Article 62, 9 pp. A MATRIX INEQUALITY FOR MÖBIUS FUNCTIONS OLIVIER BORDELLÈS AND BENOIT CLOITRE 2 ALLÉE DE LA COMBE 43000 AIGUILHE (FRANCE) [email protected] 19 RUE LOUISE MICHEL 92300 LEVALLOIS-PERRET (FRANCE) [email protected] Received 24 November, 2008; accepted 27 March, 2009 Communicated by L. Tóth ABSTRACT. The aim of this note is the study of an integer matrix whose determinant is related to the Möbius function. We derive a number-theoretic inequality involving sums of a certain class of Möbius functions and obtain a sufficient condition for the Riemann hypothesis depending on an integer triangular matrix. We also provide an alternative proof of Redheffer’s theorem based upon a LU decomposition of the Redheffer’s matrix. Key words and phrases: Determinants, Dirichlet convolution, Möbius functions, Singular values. 2000 Mathematics Subject Classification. 15A15, 11A25, 15A18, 11C20. 1. INTRODUCTION In what follows, [t] is the integer part of t and, for integers i, j > 1, we set mod(j, i) := j − i[j/i]. 1.1. Arithmetic motivation. In 1977, Redheffer [5] introduced the matrix Rn = (rij) ∈ Mn({0, 1}) defined by ( 1, if i | j or j = 1; rij = 0, otherwise and has shown that (see appendix) n X det Rn = M(n) := µ(k), k=1 where µ is the Möbius function and M is the Mertens function. This determinant is clearly related to two of the most famous problems in number theory, the Prime Number Theorem (PNT) and the Riemann Hypothesis (RH). Indeed, it is well-known that 1/2+ε PNT ⇐⇒ M(n) = o(n) and RH ⇐⇒ M(n) = Oε n 317-08 2 OLIVIER BORDELLÈS AND BENOIT CLOITRE (for any ε > 0). -
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. -
Mathematisches Forschungsinstitut Oberwolfach Analytic Number Theory
Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2019 DOI: 10.4171/OWR/2019/50 Analytic Number Theory Organized by J¨org Br¨udern, G¨ottingen Kaisa Matom¨aki, Turku Robert C. Vaughan, State College Trevor D. Wooley, West Lafayette 3 November – 9 November 2019 Abstract. Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stim- ulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions. Mathematics Subject Classification (2010): 11Bxx, 11Dxx, 11Jxx-11Pxx. Introduction by the Organizers The workshop Analytic Number Theory, organised by J¨org Br¨udern (G¨ottingen), Kaisa Matom¨aki (Turku), Robert C. Vaughan (State College) and Trevor D. Woo- ley (West Lafayette) was well attended with over 50 participants from a broad geographic spectrum, all either distinguished and leading workers in the field or very promising younger researchers. Over the last decade or so there has been considerable and surprising movement in our understanding of a whole series of questions in analytic number theory and related areas. This includes the establishment of infinitely many bounded gaps in the primes by Zhang, Maynard and Tao, significant progress on our understand- ing of the behaviour of multiplicative functions such as the M¨obius function by Matom¨aki and Radziwi l l, the work on the Vinogradov -
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]).