On a Generalization of the Lerch Zeta Function PDF Download
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Author: Dorothy Marguerite Rivera
Publisher:
ISBN:
Category : Functions, Zeta
Languages : en
Pages : 92
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Book Description
Application of a Mellin transform to a series which represents a generalization of the Lerch zeta function yields a transformation series. One obtains the asymptotic behavior of the series, together with some associated expansions and limit relations and moreover, a specialization of parameters yields several classical results.
Author: Dorothy Marguerite Rivera
Publisher:
ISBN:
Category : Functions, Zeta
Languages : en
Pages : 92
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Book Description
Application of a Mellin transform to a series which represents a generalization of the Lerch zeta function yields a transformation series. One obtains the asymptotic behavior of the series, together with some associated expansions and limit relations and moreover, a specialization of parameters yields several classical results.
Author: Antanas Laurincikas
Publisher: Springer Science & Business Media
ISBN: 9401764018
Category : Mathematics
Languages : en
Pages : 192
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Book Description
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: Hari M. Srivastava
Publisher: Springer Science & Business Media
ISBN: 9780792370543
Category : Mathematics
Languages : en
Pages : 408
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Book Description
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: Hari M. Srivastava
Publisher: Springer
ISBN: 9789401596725
Category : Mathematics
Languages : en
Pages : 0
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Book Description
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
Publisher: Elsevier
ISBN: 0123852188
Category : Mathematics
Languages : en
Pages : 675
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Book Description
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: Athanassios S. Fokas
Publisher: American Mathematical Society
ISBN: 1470450984
Category : Mathematics
Languages : en
Pages : 114
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Book Description
View the abstract.
Author: Hari M Srivastava
Publisher: Elsevier
ISBN: 0123852196
Category : Mathematics
Languages : en
Pages : 675
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Book Description
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: Anatoly A. Karatsuba
Publisher: Walter de Gruyter
ISBN: 3110886146
Category : Mathematics
Languages : en
Pages : 409
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Book Description
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: Source Wikipedia
Publisher: University-Press.org
ISBN: 9781230527567
Category :
Languages : en
Pages : 84
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Book Description
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 83. Chapters: Riemann zeta function, Dirichlet's theorem on arithmetic progressions, Generalized Riemann hypothesis, Langlands program, Dirichlet character, Polylogarithm, Weil conjectures, Dirichlet eta function, Basel problem, Hurwitz zeta function, Birch and Swinnerton-Dyer conjecture, Shimura variety, Zeta function universality, Lindelof hypothesis, Apery's constant, Zeta function regularization, Dirichlet series, Zeta constant, Euler product, Apery's theorem, Artin L-function, Dedekind zeta function, Lerch zeta function, Hecke character, Riesz function, Subgroup growth, Explicit formula, Multiplication theorem, Hilbert-Polya conjecture, Local zeta-function, Proof of the Euler product formula for the Riemann zeta function, Rankin-Selberg method, Hasse-Weil zeta function, Rational zeta series, Riemann-Siegel theta function, P-adic L-function, Dirichlet L-function, Functional equation, Riemann-Siegel formula, Special values of L-functions, Selberg class, Stark conjectures, Brumer-Stark conjecture, Clausen function, Selberg zeta function, Li's criterion, Riesz mean, Minakshisundaram-Pleijel zeta function, Multiple zeta function, Selberg's conjecture, Dirichlet beta function, Stieltjes constants, Montgomery's pair correlation conjecture, Hardy-Littlewood zeta-function conjectures, Ramanujan-Petersson conjecture, Ihara zeta function, Lefschetz zeta function, Siegel zero, Odlyzko-Schonhage algorithm, Hadjicostas's formula, Riemann Xi function, Barnes zeta function, Artin-Mazur zeta function, Prime zeta function, Beurling zeta function, Witten zeta function, Weil's criterion, Shintani zeta function, Fekete polynomial, Eichler-Shimura congruence relation, Chowla-Mordell theorem, ZetaGrid, Equivariant L-function, Goss zeta function, Grand Riemann hypothesis, Matsumoto zeta function, Airy zeta function.
Author: Harendra Singh
Publisher: CRC Press
ISBN: 1000899756
Category : Technology & Engineering
Languages : en
Pages : 315
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Book Description
Special functions play a very important role in solving various families of ordinary and partial differential equations as well as their fractional-order analogs, which model real-life situations. Owing to the non-local nature and memory effect, fractional calculus is capable of modeling many situations which arise in engineering. This book includes a collection of related topics associated with such equations and their relevance and significance in engineering. Special Functions in Fractional Calculus and Engineering highlights the significance and applicability of special functions in solving fractional-order differential equations with engineering applications. This book focuses on the non-local nature and memory effect of fractional calculus in modeling relevant to engineering science and covers a variety of important and useful methods using special functions for solving various types of fractional-order models relevant to engineering science. This book goes on to illustrate the applicability and usefulness of special functions by justifying their numerous and widespread occurrences in the solution of fractional-order differential, integral, and integrodifferential equations. This book holds a wide variety of interconnected fundamental and advanced topics with interdisciplinary applications that combine applied mathematics and engineering sciences, which are useful to graduate students, Ph.D. scholars, researchers, and educators interested in special functions, fractional calculus, mathematical modeling, and engineering.
