The Theory Of The Riemann Zeta Function

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 : 44,8 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.

Author : Titchmarch E. C.
Publisher : Unknown
Page : 128 pages
File Size : 53,9 Mb
Release : 1992
Category : Electronic
ISBN : OCLC:786156446

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Theory of Functions by Titchmarch E. C. Pdf

Author : Yoichi Motohashi
Publisher : Cambridge University Press
Page : 246 pages
File Size : 53,9 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 : S. J. Patterson
Publisher : Cambridge University Press
Page : 176 pages
File Size : 40,8 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 : Anonim
Publisher : Unknown
Page : 346 pages
File Size : 40,9 Mb
Release : 1951
Category : Electronic
ISBN : OCLC:474173702

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The Theory of the Riemann Zeta-function by Anonim Pdf

Author : Aleksandar Ivic
Publisher : Courier Corporation
Page : 548 pages
File Size : 43,9 Mb
Release : 2012-07-12
Category : Mathematics
ISBN : 9780486140049

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The Riemann Zeta-Function by Aleksandar Ivic Pdf

This text covers exponential integrals and sums, 4th power moment, zero-free region, mean value estimates over short intervals, higher power moments, omega results, zeros on the critical line, zero-density estimates, and more. 1985 edition.

Author : Edward Charles Titchmarsh
Publisher : Unknown
Page : 454 pages
File Size : 46,8 Mb
Release : 1960
Category : Functions, Zeta
ISBN : OCLC:223795733

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The Theory of the Riemann Zeta-function by Edward Charles Titchmarsh Pdf

Author : Hugh Montgomery,Ashkan Nikeghbali,Michael Th. Rassias
Publisher : Springer
Page : 298 pages
File Size : 42,6 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 : H. Iwaniec
Publisher : American Mathematical Society
Page : 130 pages
File Size : 46,9 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 : 42,5 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 : John Coates,A. Raghuram,Anupam Saikia,R. Sujatha
Publisher : Cambridge University Press
Page : 317 pages
File Size : 49,8 Mb
Release : 2015-03-13
Category : Mathematics
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 : 43,9 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 : Anatoly A. Karatsuba,S. M. Voronin
Publisher : Walter de Gruyter
Page : 409 pages
File Size : 45,5 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 : Peter B. Borwein
Publisher : Springer Science & Business Media
Page : 543 pages
File Size : 43,6 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 : Jürgen Fischer
Publisher : Springer
Page : 188 pages
File Size : 43,7 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540393313

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An Approach to the Selberg Trace Formula via the Selberg Zeta-Function by Jürgen Fischer Pdf

The Notes give a direct approach to the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) acting on the upper half-plane. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic Laplacian in order to arrive at the logarithmic derivative of the Selberg zeta-function. Previous knowledge of the Selberg trace formula is not assumed. The theory is developed for arbitrary real weights and for arbitrary multiplier systems permitting an approach to known results on classical automorphic forms without the Riemann-Roch theorem. The author's discussion of the Selberg trace formula stresses the analogy with the Riemann zeta-function. For example, the canonical factorization theorem involves an analogue of the Euler constant. Finally the general Selberg trace formula is deduced easily from the properties of the Selberg zeta-function: this is similar to the procedure in analytic number theory where the explicit formulae are deduced from the properties of the Riemann zeta-function. Apart from the basic spectral theory of the Laplacian for cofinite groups the book is self-contained and will be useful as a quick approach to the Selberg zeta-function and the Selberg trace formula.