P Adic Numbers P Adic Analysis And Zeta Functions

Author : NEAL Koblitz
Publisher : Springer Science & Business Media
Page : 134 pages
File Size : 53,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468400472

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p-adic Numbers, p-adic Analysis, and Zeta-Functions by NEAL Koblitz Pdf

These lecture notes are intended as an introduction to p-adic analysis on the elementary level. For this reason they presuppose as little background as possi ble. Besides about three semesters of calculus, I presume some slight exposure to more abstract mathematics, to the extent that the student won't have an adverse reaction to matrices with entries in a field other than the real numbers, field extensions of the rational numbers, or the notion of a continuous map of topolog ical spaces. The purpose of this book is twofold: to develop some basic ideas of p-adic analysis, and to present two striking applications which, it is hoped, can be as effective pedagogically as they were historically in stimulating interest in the field. The first of these applications is presented in Chapter II, since it only requires the most elementary properties of Q ; this is Mazur's construction by p means of p-adic integration of the Kubota-Leopoldtp-adic zeta-function, which "p-adically interpolates" the values of the Riemann zeta-function at the negative odd integers. My treatment is based on Mazur's Bourbaki notes (unpublished).

Author : Alain M. Robert
Publisher : Springer Science & Business Media
Page : 451 pages
File Size : 54,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475732542

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A Course in p-adic Analysis by Alain M. Robert Pdf

Discovered at the turn of the 20th century, p-adic numbers are frequently used by mathematicians and physicists. This text is a self-contained presentation of basic p-adic analysis with a focus on analytic topics. It offers many features rarely treated in introductory p-adic texts such as topological models of p-adic spaces inside Euclidian space, a special case of Hazewinkel’s functional equation lemma, and a treatment of analytic elements.

Author : Paul Monsky
Publisher : Unknown
Page : 136 pages
File Size : 48,9 Mb
Release : 1970
Category : Functions, Zeta
ISBN : UOM:39015017329197

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P-Adic Analysis and Zeta Functions by Paul Monsky Pdf

Author : Svetlana Katok
Publisher : American Mathematical Soc.
Page : 152 pages
File Size : 50,8 Mb
Release : 2007
Category : Mathematics
ISBN : 9780821842201

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P-adic Analysis Compared with Real by Svetlana Katok Pdf

The book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study.

Author : Neal Koblitz
Publisher : Cambridge University Press
Page : 171 pages
File Size : 45,7 Mb
Release : 1980-11-28
Category : Mathematics
ISBN : 9780521280600

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P-adic Analysis by Neal Koblitz Pdf

An introduction to recent work in the theory of numbers and its interrelation with algebraic geometry and analysis.

Author : Anatoly A. Karatsuba,S. M. Voronin
Publisher : Walter de Gruyter
Page : 409 pages
File Size : 41,7 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 : Vasili? Sergeevich Vladimirov,I. V. Volovich,E. I. Zelenov
Publisher : World Scientific
Page : 350 pages
File Size : 40,5 Mb
Release : 1994
Category : Science
ISBN : 9810208804

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P-adic Analysis and Mathematical Physics by Vasili? Sergeevich Vladimirov,I. V. Volovich,E. I. Zelenov Pdf

p-adic numbers play a very important role in modern number theory, algebraic geometry and representation theory. Lately p-adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the non-Archimedean geometry of space-time at small distances.This is the first book to deal with applications of p-adic numbers in theoretical and mathematical physics. It gives an elementary and thoroughly written introduction to p-adic numbers and p-adic analysis with great numbers of examples as well as applications of p-adic numbers in classical mechanics, dynamical systems, quantum mechanics, statistical physics, quantum field theory and string theory.

Author : M. Ram Murty
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 47,8 Mb
Release : 2009-02-09
Category : Mathematics
ISBN : 9780821847749

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Introduction to $p$-adic Analytic Number Theory by M. Ram Murty Pdf

This book is an elementary introduction to $p$-adic analysis from the number theory perspective. With over 100 exercises included, it will acquaint the non-expert to the basic ideas of the theory and encourage the novice to enter this fertile field of research. The main focus of the book is the study of $p$-adic $L$-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. These congruences are then used to construct the $p$-adic analog of the Riemann zeta function and $p$-adic analogs of Dirichlet's $L$-functions. Featured is a chapter on how to apply the theory of Newton polygons to determine Galois groups of polynomials over the rational number field. As motivation for further study, the final chapter introduces Iwasawa theory.

Author : Fernando Q. Gouvea
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 48,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662222782

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p-adic Numbers by Fernando Q. Gouvea Pdf

p-adic numbers are of great theoretical importance in number theory, since they allow the use of the language of analysis to study problems relating toprime numbers and diophantine equations. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own (a large appendix with hints on the problem is included). Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.

Author : Neal Koblitz
Publisher : Springer Science & Business Media
Page : 163 pages
File Size : 41,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461211129

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p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz Pdf

The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper treatment of p-adic functions in Ch. 4 to include the Iwasawa logarithm and the p-adic gamma-function, the rearrangement and addition of some exercises, the inclusion of an extensive appendix of answers and hints to the exercises, as well as numerous clarifications.

Author : Maruti Ram Murty
Publisher : Unknown
Page : 149 pages
File Size : 51,9 Mb
Release : 2002
Category : Number theory
ISBN : 1470417421

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Introduction to $p$-adic Analytic Number Theory by Maruti Ram Murty Pdf

This book is an elementary introduction to p-adic analysis from the number theory perspective. With over 100 exercises included, it will acquaint the non-expert to the basic ideas of the theory and encourage the novice to enter this fertile field of research. The main focus of the book is the study of p-adic L-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. These congruences are then used to construct the p-adic analog of the Riemann zeta function and p-adic analogs of Dirichlet's L-functi.

Author : Fernando Q. Gouvêa
Publisher : Springer Nature
Page : 366 pages
File Size : 47,6 Mb
Release : 2020-06-19
Category : Mathematics
ISBN : 9783030472955

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p-adic Numbers by Fernando Q. Gouvêa Pdf

There are numbers of all kinds: rational, real, complex, p-adic. The p-adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. This elementary introduction offers a broad understanding of p-adic numbers. From the reviews: "It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." --THE MATHEMATICAL GAZETTE

Author : Bernard Dwork,Giovanni Gerotto,Francis J. Sullivan
Publisher : Princeton University Press
Page : 349 pages
File Size : 55,9 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882540

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An Introduction to G-Functions. (AM-133), Volume 133 by Bernard Dwork,Giovanni Gerotto,Francis J. Sullivan Pdf

Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, André, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

Author : Alain Escassut
Publisher : World Scientific
Page : 559 pages
File Size : 43,5 Mb
Release : 2015-11-27
Category : Mathematics
ISBN : 9789814730112

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Value Distribution in P-Adic Analysis by Alain Escassut Pdf

"The book first explains the main properties of analytic functions in order to use them in the study of various problems in p-adic value distribution. Certain properties of p-adic transcendental numbers are examined such as order and type of transcendence, with problems on p-adic exponentials. Lazard's problem for analytic functions inside a disk is explained. P-adic meromorphics are studied. Sets of range uniqueness in a p-adic field are examined. The ultrametric Corona problem is studied. Injective analytic elements are characterized. The p-adic Nevanlinna theory is described and many applications are given: p-adic Hayman conjecture, Picard's values for derivatives, small functions, branched values, growth of entire functions, problems of uniqueness, URSCM and URSIM, functions of uniqueness, sharing value problems, Nevanlinna theory in characteristic p>0, p-adic Yosida's equation."--

Author : John Coates,R. Sujatha
Publisher : Springer Science & Business Media
Page : 120 pages
File Size : 53,8 Mb
Release : 2006-10-03
Category : Mathematics
ISBN : 9783540330691

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Cyclotomic Fields and Zeta Values by John Coates,R. Sujatha Pdf

Written by two leading workers in the field, this brief but elegant book presents in full detail the simplest proof of the "main conjecture" for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. From the reviews: "The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details." --ZENTRALBLATT MATH