The Riemann Zeta Function

Author : H. Iwaniec
Publisher : American Mathematical Society
Page : 119 pages
File Size : 47,7 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9781470418519

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Lectures on the Riemann Zeta Function by H. Iwaniec Pdf

The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

Author : Harold M. Edwards
Publisher : Courier Corporation
Page : 338 pages
File Size : 43,9 Mb
Release : 2001-01-01
Category : Mathematics
ISBN : 0486417409

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Riemann's Zeta Function by Harold M. Edwards Pdf

Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.

Author : E. C. Titchmarsh
Publisher : Unknown
Page : 452 pages
File Size : 45,9 Mb
Release : 1964
Category : Calculus
ISBN : OCLC:1068462371

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The Theory of Functions by E. C. Titchmarsh Pdf

Author : Hugh Montgomery,Ashkan Nikeghbali,Michael Th. Rassias
Publisher : Springer
Page : 298 pages
File Size : 40,6 Mb
Release : 2017-09-11
Category : Mathematics
ISBN : 9783319599694

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Exploring the Riemann Zeta Function by Hugh Montgomery,Ashkan Nikeghbali,Michael Th. Rassias Pdf

Exploring the Riemann Zeta Function: 190 years from Riemann's Birth presents a collection of chapters contributed by eminent experts devoted to the Riemann Zeta Function, its generalizations, and their various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis, Probability Theory, and related subjects. The book focuses on both old and new results towards the solution of long-standing problems as well as it features some key historical remarks. The purpose of this volume is to present in a unified way broad and deep areas of research in a self-contained manner. It will be particularly useful for graduate courses and seminars as well as it will make an excellent reference tool for graduate students and researchers in Mathematics, Mathematical Physics, Engineering and Cryptography.

Author : John Coates,A. Raghuram,Anupam Saikia,R. Sujatha
Publisher : Cambridge University Press
Page : 317 pages
File Size : 44,7 Mb
Release : 2015-03-13
Category : Literary Criticism
ISBN : 9781107492967

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The Bloch-Kato Conjecture for the Riemann Zeta Function by John Coates,A. Raghuram,Anupam Saikia,R. Sujatha Pdf

A graduate-level account of an important recent result concerning the Riemann zeta function.

Author : Antanas Laurincikas
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 50,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401720915

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Limit Theorems for the Riemann Zeta-Function by Antanas Laurincikas Pdf

The subject of this book is probabilistic number theory. In a wide sense probabilistic number theory is part of the analytic number theory, where the methods and ideas of probability theory are used to study the distribution of values of arithmetic objects. This is usually complicated, as it is difficult to say anything about their concrete values. This is why the following problem is usually investigated: given some set, how often do values of an arithmetic object get into this set? It turns out that this frequency follows strict mathematical laws. Here we discover an analogy with quantum mechanics where it is impossible to describe the chaotic behaviour of one particle, but that large numbers of particles obey statistical laws. The objects of investigation of this book are Dirichlet series, and, as the title shows, the main attention is devoted to the Riemann zeta-function. In studying the distribution of values of Dirichlet series the weak convergence of probability measures on different spaces (one of the principle asymptotic probability theory methods) is used. The application of this method was launched by H. Bohr in the third decade of this century and it was implemented in his works together with B. Jessen. Further development of this idea was made in the papers of B. Jessen and A. Wintner, V. Borchsenius and B.

Author : Yoichi Motohashi
Publisher : Cambridge University Press
Page : 246 pages
File Size : 48,7 Mb
Release : 1997-09-11
Category : Mathematics
ISBN : 9780521445207

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Spectral Theory of the Riemann Zeta-Function by Yoichi Motohashi Pdf

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Author : Samuel W. Gilbert
Publisher : Riemann hypothesis
Page : 160 pages
File Size : 41,8 Mb
Release : 2009
Category : Mathematics
ISBN : 143921638X

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The Riemann Hypothesis and the Roots of the Riemann Zeta Function by Samuel W. Gilbert Pdf

The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.

Author : Anatoly A. Karatsuba,S. M. Voronin
Publisher : Walter de Gruyter
Page : 409 pages
File Size : 45,6 Mb
Release : 2011-05-03
Category : Mathematics
ISBN : 9783110886146

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The Riemann Zeta-Function by Anatoly A. Karatsuba,S. M. Voronin Pdf

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Author : Barry Mazur,William Stein
Publisher : Cambridge University Press
Page : 155 pages
File Size : 50,7 Mb
Release : 2016-04-11
Category : Mathematics
ISBN : 9781107101920

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Prime Numbers and the Riemann Hypothesis by Barry Mazur,William Stein Pdf

This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.

Author : Aleksandar Ivic
Publisher : Courier Corporation
Page : 548 pages
File Size : 41,9 Mb
Release : 2012-07-12
Category : Mathematics
ISBN : 9780486140049

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The Riemann Zeta-Function by Aleksandar Ivic Pdf

"A thorough and easily accessible account."—MathSciNet, Mathematical Reviews on the Web, American Mathematical Society. This extensive survey presents a comprehensive and coherent account of Riemann zeta-function theory and applications. Starting with elementary theory, it examines exponential integrals and exponential sums, the Voronoi summation formula, the approximate functional equation, the fourth power moment, the zero-free region, mean value estimates over short intervals, higher power moments, and omega results. Additional topics include zeros on the critical line, zero-density estimates, the distribution of primes, the Dirichlet divisor problem and various other divisor problems, and Atkinson's formula for the mean square. End-of-chapter notes supply the history of each chapter's topic and allude to related results not covered by the book. 1985 edition.

Author : Peter B. Borwein
Publisher : Springer Science & Business Media
Page : 543 pages
File Size : 47,9 Mb
Release : 2008
Category : Mathematics
ISBN : 9780387721255

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The Riemann Hypothesis by Peter B. Borwein Pdf

The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors." The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

Author : S. J. Patterson
Publisher : Cambridge University Press
Page : 176 pages
File Size : 52,8 Mb
Release : 1995-02-02
Category : Mathematics
ISBN : 0521499054

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An Introduction to the Theory of the Riemann Zeta-Function by S. J. Patterson Pdf

An introduction to the analytic techniques used in the investigation of zeta functions through the example of the Riemann zeta function. It emphasizes central ideas of broad application, avoiding technical results and the customary function-theoretic appro

Author : Antanas Laurincikas,Ramunas Garunkstis
Publisher : Springer Science & Business Media
Page : 192 pages
File Size : 41,9 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9789401764018

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The Lerch zeta-function by Antanas Laurincikas,Ramunas Garunkstis Pdf

The Lerch zeta-function is the first monograph on this topic, which is a generalization of the classic Riemann, and Hurwitz zeta-functions. Although analytic results have been presented previously in various monographs on zeta-functions, this is the first book containing both analytic and probability theory of Lerch zeta-functions. The book starts with classical analytical theory (Euler gamma-functions, functional equation, mean square). The majority of the presented results are new: on approximate functional equations and its applications and on zero distribution (zero-free regions, number of nontrivial zeros etc). Special attention is given to limit theorems in the sense of the weak convergence of probability measures for the Lerch zeta-function. From limit theorems in the space of analytic functions the universitality and functional independence is derived. In this respect the book continues the research of the first author presented in the monograph Limit Theorems for the Riemann zeta-function. This book will be useful to researchers and graduate students working in analytic and probabilistic number theory, and can also be used as a textbook for postgraduate students.

Author : Late Savilian Professor of Geometry E C Titchmarsh,Edward Charles Titchmarsh,Titchmarsh,D. R. Heath-Brown,Titchmarsh, Edward Charles Titchmarsh
Publisher : Oxford University Press
Page : 428 pages
File Size : 42,5 Mb
Release : 1986
Category : Mathematics
ISBN : 0198533691

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The Theory of the Riemann Zeta-function by Late Savilian Professor of Geometry E C Titchmarsh,Edward Charles Titchmarsh,Titchmarsh,D. R. Heath-Brown,Titchmarsh, Edward Charles Titchmarsh Pdf

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.