From Arithmetic To Zeta Functions

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Dynamical, Spectral, and Arithmetic Zeta Functions

Author : Michel Laurent Lapidus
Publisher : American Mathematical Soc.
ISBN 13 : 0821820796
Total Pages : 210 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Dynamical, Spectral, and Arithmetic Zeta Functions by : Michel Laurent Lapidus

Download or read book Dynamical, Spectral, and Arithmetic Zeta Functions written by Michel Laurent Lapidus and published by American Mathematical Soc.. This book was released on 2001 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results. This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function. Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.

Zeta Functions in Algebra and Geometry

Author : Antonio Campillo
Publisher : American Mathematical Soc.
ISBN 13 : 0821869000
Total Pages : 344 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Zeta Functions in Algebra and Geometry by : Antonio Campillo

Download or read book Zeta Functions in Algebra and Geometry written by Antonio Campillo and published by American Mathematical Soc.. This book was released on 2012 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume contains the proceedings of the ``Second International Workshop on Zeta Functions in Algebra and Geometry'' held May 3-7, 2010 at the Universitat de les Illes Balears, Palma de Mallorca, Spain. Zeta functions can be naturally attached to several mathematical objects, including fields, groups, and algebras. The conference focused on the following topics: arithmetic and geometric aspects of local, topological, and motivic zeta functions, Poincare series of valuations, zeta functions of groups, rings, and representations, prehomogeneous vector spaces and their zeta functions, and height zeta functions. This book is published in cooperation with Real Sociedad Matematica Espanola (RSME).

From Arithmetic to Zeta-Functions

Author : Jürgen Sander
Publisher : Springer
ISBN 13 : 3319282034
Total Pages : 552 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis From Arithmetic to Zeta-Functions by : Jürgen Sander

Download or read book From Arithmetic to Zeta-Functions written by Jürgen Sander and published by Springer. This book was released on 2016-12-29 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Zeta Functions of Groups and Rings

Author : Marcus du Sautoy
Publisher : Springer Science & Business Media
ISBN 13 : 354074701X
Total Pages : 217 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Zeta Functions of Groups and Rings by : Marcus du Sautoy

Download or read book Zeta Functions of Groups and Rings written by Marcus du Sautoy and published by Springer Science & Business Media. This book was released on 2008 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic behaviour of these functions together with an investigation of functional equations. Many important examples of zeta functions are calculated and recorded providing an important data base of explicit examples and methods for calculation.

Limit Theorems for the Riemann Zeta-Function

Author : Antanas Laurincikas
Publisher : Springer Science & Business Media
ISBN 13 : 9401720916
Total Pages : 316 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis Limit Theorems for the Riemann Zeta-Function by : Antanas Laurincikas

Download or read book Limit Theorems for the Riemann Zeta-Function written by Antanas Laurincikas and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves

Author : Spencer J. Bloch
Publisher : American Mathematical Soc.
ISBN 13 : 0821829734
Total Pages : 97 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves by : Spencer J. Bloch

Download or read book Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves written by Spencer J. Bloch and published by American Mathematical Soc.. This book was released on 2000 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: This brief hardcover is a classic title covering a renowned series of lectures. A mathematical jewel.

Zeta Functions over Zeros of Zeta Functions

Author : André Voros
Publisher : Springer Science & Business Media
ISBN 13 : 3642052037
Total Pages : 163 pages
Book Rating : 4.6/5 (42 download)

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Book Synopsis Zeta Functions over Zeros of Zeta Functions by : André Voros

Download or read book Zeta Functions over Zeros of Zeta Functions written by André Voros and published by Springer Science & Business Media. This book was released on 2009-11-21 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this text, the famous zeros of the Riemann zeta function and its generalizations (L-functions, Dedekind and Selberg zeta functions)are analyzed through several zeta functions built over those zeros.

Lectures on the Riemann Zeta Function

Author : H. Iwaniec
Publisher : American Mathematical Society
ISBN 13 : 1470418517
Total Pages : 119 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Lectures on the Riemann Zeta Function by : H. Iwaniec

Download or read book Lectures on the Riemann Zeta Function written by H. Iwaniec and published by American Mathematical Society. This book was released on 2014-10-07 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

The Bloch–Kato Conjecture for the Riemann Zeta Function

Author : John Coates
Publisher : Cambridge University Press
ISBN 13 : 1316241300
Total Pages : pages
Book Rating : 4.3/5 (162 download)

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Book Synopsis The Bloch–Kato Conjecture for the Riemann Zeta Function by : John Coates

Download or read book The Bloch–Kato Conjecture for the Riemann Zeta Function written by John Coates and published by Cambridge University Press. This book was released on 2015-03-19 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

The Theory of the Riemann Zeta-function

Author : Late Savilian Professor of Geometry E C Titchmarsh
Publisher : Oxford University Press
ISBN 13 : 9780198533696
Total Pages : 428 pages
Book Rating : 4.5/5 (336 download)

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Book Synopsis The Theory of the Riemann Zeta-function by : Late Savilian Professor of Geometry E C Titchmarsh

Download or read book The Theory of the Riemann Zeta-function written by Late Savilian Professor of Geometry E C Titchmarsh and published by Oxford University Press. This book was released on 1986 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Zeta Functions, Topology and Quantum Physics

Author : Takashi Aoki
Publisher : Springer Science & Business Media
ISBN 13 : 0387249818
Total Pages : 219 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Zeta Functions, Topology and Quantum Physics by : Takashi Aoki

Download or read book Zeta Functions, Topology and Quantum Physics written by Takashi Aoki and published by Springer Science & Business Media. This book was released on 2008-05-10 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains papers by invited speakers of the symposium "Zeta Functions, Topology and Quantum Physics" held at Kinki U- versity in Osaka, Japan, during the period of March 3-6, 2003. The aims of this symposium were to establish mutual understanding and to exchange ideas among researchers working in various fields which have relation to zeta functions and zeta values. We are very happy to add this volume to the series Developments in Mathematics from Springer. In this respect, Professor Krishnaswami Alladi helped us a lot by showing his keen and enthusiastic interest in publishing this volume and by contributing his paper with Alexander Berkovich. We gratefully acknowledge financial support from Kinki University. We would like to thank Professor Megumu Munakata, Vice-Rector of Kinki University, and Professor Nobuki Kawashima, Director of School of Interdisciplinary Studies of Science and Engineering, Kinki Univ- sity, for their interest and support. We also thank John Martindale of Springer for his excellent editorial work.

History of Zeta Functions

Author : Robert Spira
Publisher :
ISBN 13 :
Total Pages : 424 pages
Book Rating : 4.E/5 ( download)

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Book Synopsis History of Zeta Functions by : Robert Spira

Download or read book History of Zeta Functions written by Robert Spira and published by . This book was released on 1999 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Lerch zeta-function

Author : Antanas Laurincikas
Publisher : Springer Science & Business Media
ISBN 13 : 9401764018
Total Pages : 192 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis The Lerch zeta-function by : Antanas Laurincikas

Download or read book The Lerch zeta-function written by Antanas Laurincikas and published by Springer Science & Business Media. This book was released on 2013-12-11 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Zeta Functions Of Reductive Groups And Their Zeros

Author : Weng Lin
Publisher : World Scientific
ISBN 13 : 9813230665
Total Pages : 556 pages
Book Rating : 4.8/5 (132 download)

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Book Synopsis Zeta Functions Of Reductive Groups And Their Zeros by : Weng Lin

Download or read book Zeta Functions Of Reductive Groups And Their Zeros written by Weng Lin and published by World Scientific. This book was released on 2018-02-07 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a systematic account of several breakthroughs in the modern theory of zeta functions. It contains two different approaches to introduce and study genuine zeta functions for reductive groups (and their maximal parabolic subgroups) defined over number fields. Namely, the geometric one, built up from stability of principal lattices and an arithmetic cohomology theory, and the analytic one, from Langlands' theory of Eisenstein systems and some techniques used in trace formula, respectively. Apparently different, they are unified via a Lafforgue type relation between Arthur's analytic truncations and parabolic reductions of Harder–Narasimhan and Atiyah–Bott. Dominated by the stability condition and/or the Lie structures embedded in, these zeta functions have a standard form of the functional equation, admit much more refined symmetric structures, and most surprisingly, satisfy a weak Riemann hypothesis. In addition, two levels of the distributions for their zeros are exposed, i.e. a classical one giving the Dirac symbol, and a secondary one conjecturally related to GUE. This book is written not only for experts, but for graduate students as well. For example, it offers a summary of basic theories on Eisenstein series and stability of lattices and arithmetic principal torsors. The second part on rank two zeta functions can be used as an introduction course, containing a Siegel type treatment of cusps and fundamental domains, and an elementary approach to the trace formula involved. Being in the junctions of several branches and advanced topics of mathematics, these works are very complicated, the results are fundamental, and the theory exposes a fertile area for further research. Contents: Non-Abelian Zeta Functions Rank Two Zeta Functions Eisenstein Periods and Multiple L-Functions Zeta Functions for Reductive Groups Algebraic, Analytic Structures and Rieman Hypothesis Geometric Structures and Riemann Hypothesis Five Essays on Arithmetic Cohomology Readership: Graduate students and researchers in the theory of zeta functions. Keywords: Zeta Function;Riemann Hypothesis;Stability;Lattice;Fundamental Domain;Reductive Group;Root System;Eisenstein Series;Truncation;Arithmetic Principal Torsor;Adelic CohomologyReview: Key Features: Genuine zeta functions for reductive groups over number fields are introduced and studied systematically, based on (i) fine parabolic structures and Lie structures involved, (ii) a new stability theory for arithmetic principal torsors over number fields, and (iii) trace formula via a geometric understanding of Arthur's analytic truncations For the first time in history, we prove a weak Riemann hypothesis for zeta functions of reductive groups defined over number fields Not only the theory is explained, but the process of building the theory is elaborated in great detail

Automorphic Forms, Representation Theory and Arithmetic

Author : S. Gelbart
Publisher : Springer
ISBN 13 : 3662007347
Total Pages : 358 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Automorphic Forms, Representation Theory and Arithmetic by : S. Gelbart

Download or read book Automorphic Forms, Representation Theory and Arithmetic written by S. Gelbart and published by Springer. This book was released on 2013-12-01 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: International Colloquium an Automorphic Forms, Representation Theory and Arithmetic. Published for the Tata Institute of Fundamental Research, Bombay

Bernoulli Numbers and Zeta Functions

Author : Tsuneo Arakawa
Publisher : Springer
ISBN 13 : 4431549196
Total Pages : 274 pages
Book Rating : 4.4/5 (315 download)

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Book Synopsis Bernoulli Numbers and Zeta Functions by : Tsuneo Arakawa

Download or read book Bernoulli Numbers and Zeta Functions written by Tsuneo Arakawa and published by Springer. This book was released on 2014-07-11 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the doub le zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.

Introduction to the Arithmetic Theory of Automorphic Functions

Author : Gorō Shimura
Publisher : Princeton University Press
ISBN 13 : 9780691080925
Total Pages : 292 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Introduction to the Arithmetic Theory of Automorphic Functions by : Gorō Shimura

Download or read book Introduction to the Arithmetic Theory of Automorphic Functions written by Gorō Shimura and published by Princeton University Press. This book was released on 1971-08-21 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.