The Lerch Zeta Function

Author : Antanas Laurincikas,Ramunas Garunkstis
Publisher : Springer Science & Business Media
Page : 192 pages
File Size : 50,9 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 : Vivek V. Rane
Publisher : Unknown
Page : 128 pages
File Size : 55,7 Mb
Release : 2015
Category : Electronic
ISBN : 8184874464

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Theory of the Hurwitz Zeta Function of the Second Variable by Vivek V. Rane Pdf

Author : Hari M. Srivastava,Junesang Choi
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 46,7 Mb
Release : 2001
Category : Mathematics
ISBN : 0792370546

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Series Associated With the Zeta and Related Functions by Hari M. Srivastava,Junesang Choi Pdf

In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s,a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions.

Author : H. M. Srivastava,Junesang Choi
Publisher : Elsevier
Page : 675 pages
File Size : 45,5 Mb
Release : 2011-10-25
Category : Mathematics
ISBN : 9780123852182

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Zeta and Q-Zeta Functions and Associated Series and Integrals by H. M. Srivastava,Junesang Choi Pdf

Zeta and q-Zeta Functions and Associated Series and Integrals is a thoroughly revised, enlarged and updated version of Series Associated with the Zeta and Related Functions. Many of the chapters and sections of the book have been significantly modified or rewritten, and a new chapter on the theory and applications of the basic (or q-) extensions of various special functions is included. This book will be invaluable because it covers not only detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions, but stimulating historical accounts of a large number of problems and well-classified tables of series and integrals. Detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions

Author : Hari M. Srivastava,Junesang Choi
Publisher : Springer
Page : 0 pages
File Size : 55,9 Mb
Release : 2001
Category : Mathematics
ISBN : 9401596727

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Series Associated with the Zeta and Related Functions by Hari M. Srivastava,Junesang Choi Pdf

In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s,a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions.

Author : Anatoly A. Karatsuba,S. M. Voronin
Publisher : Walter de Gruyter
Page : 409 pages
File Size : 46,5 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 : I. S. Gradshteyn,I. M. Ryzhik
Publisher : Academic Press
Page : 1206 pages
File Size : 50,6 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483265643

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Table of Integrals, Series, and Products by I. S. Gradshteyn,I. M. Ryzhik Pdf

Table of Integrals, Series, and Products provides information pertinent to the fundamental aspects of integrals, series, and products. This book provides a comprehensive table of integrals. Organized into 17 chapters, this book begins with an overview of elementary functions and discusses the power of binomials, the exponential function, the logarithm, the hyperbolic function, and the inverse trigonometric function. This text then presents some basic results on vector operators and coordinate systems that are likely to be useful during the formulation of many problems. Other chapters consider inequalities that range from basic algebraic and functional inequalities to integral inequalities and fundamental oscillation and comparison theorems for ordinary differential equations. This book discusses as well the important part played by integral transforms. The final chapter deals with Fourier and Laplace transforms that provides so much information about other integrals. This book is a valuable resource for mathematicians, engineers, scientists, and research workers.

Author : Markus Szymon Fraczek
Publisher : Springer
Page : 354 pages
File Size : 54,6 Mb
Release : 2017-05-11
Category : Mathematics
ISBN : 9783319512969

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Selberg Zeta Functions and Transfer Operators by Markus Szymon Fraczek Pdf

This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic Laplacians under both singular and non-singular perturbations. Areas in which the theory has not yet been sufficiently developed, such as the spectral theory of transfer operators or the singular perturbation theory of hyperbolic Laplacians, will profit from the numerical experiments discussed in this book. Detailed descriptions of numerical approaches to the spectra and eigenfunctions of transfer operators and to computations of Selberg zeta functions will be of value to researchers active in analysis, while those researchers focusing more on numerical aspects will benefit from discussions of the analytic theory, in particular those concerning the transfer operator method and the spectral theory of hyperbolic spaces.

Author : Jr̲n Steuding
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 45,8 Mb
Release : 2007-06-06
Category : Mathematics
ISBN : 9783540265269

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Value-Distribution of L-Functions by Jr̲n Steuding Pdf

These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.

Author : Athanassios S. Fokas,Jonatan Lenells
Publisher : American Mathematical Society
Page : 114 pages
File Size : 55,5 Mb
Release : 2022-02-02
Category : Mathematics
ISBN : 9781470450984

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On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function by Athanassios S. Fokas,Jonatan Lenells Pdf

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Author : Jürgen Sander,Jörn Steuding,Rasa Steuding
Publisher : Springer
Page : 552 pages
File Size : 45,9 Mb
Release : 2016-12-29
Category : Mathematics
ISBN : 9783319282039

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From Arithmetic to Zeta-Functions by Jürgen Sander,Jörn Steuding,Rasa Steuding Pdf

This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.

Author : Robert Spira
Publisher : Unknown
Page : 424 pages
File Size : 51,5 Mb
Release : 1999
Category : Functions, Zeta
ISBN : CORNELL:31924086163080

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History of Zeta Functions by Robert Spira Pdf

Author : Hidehiko Mishou,Takashi Nakamura (Mathematician),Masatoshi Suzuki,Yumiko Umegaki
Publisher : Advanced Studies in Pure Mathe
Page : 0 pages
File Size : 49,7 Mb
Release : 2020
Category : Mathematics
ISBN : 4864970882

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Various Aspects of Multiple Zeta Functions by Hidehiko Mishou,Takashi Nakamura (Mathematician),Masatoshi Suzuki,Yumiko Umegaki Pdf

This volume is the proceedings of the international conference 'Various Aspects of Multiple Zeta Functions' in honor of Professor Kohji Matsumoto's 60th birthday held at Nagoya University, Japan, during August 21 to 25, 2017.The present volume consists of 15 research papers on various recent topics about multiple zeta-functions, which include not only actually multivariate cases but also single-variable cases, additive and multiplicative number theory, and poly-Bernoulli numbers and polynomials.The editors believe that this volume represents the major part of the contributions presented in the conference, and hope that the volume is useful for all researchers and students who are interested in this fruitful research field.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

Author : Aleksandar Ivic
Publisher : Courier Corporation
Page : 548 pages
File Size : 42,8 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 : Frédéric Fauvet,Dominique Manchon,Stefano Marmi,David Sauzin
Publisher : Springer
Page : 384 pages
File Size : 40,7 Mb
Release : 2017-11-17
Category : Science
ISBN : 9788876426131

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Resurgence, Physics and Numbers by Frédéric Fauvet,Dominique Manchon,Stefano Marmi,David Sauzin Pdf

This book is issued from a conference around resurgent functions in Physics and multiple zetavalues, which was held at the Centro di Ricerca Matematica Ennio de Giorgi in Pisa, on May 18-22, 2015. This meeting originally stemmed from the impressive upsurge of interest for Jean Ecalle's alien calculus in Physics, in the last years – a trend that has considerably developed since then. The volume contains both original research papers and surveys, by leading experts in the field, reflecting the themes that were tackled at this event: Stokes phenomenon and resurgence, in various mathematical and physical contexts but also related constructions in algebraic combinatorics and results concerning numbers, specifically multiple zetavalues.