數論中的模函數和狄利克萊級數

前言

　　This iS the second volume of a 2.volume textbook*which evolved from a course（Mathematics 160）offered at the Ca“fornia Institute of Technology during the last 25 years.　　The second volume presupposes a background in number theory com. parable tO that provided in the first volume。together with a knowledge of the basic concepts of complex analysis.　　Most of the present volume iS devoted to elliptic functions and modular functions with some of their number.theoretic applications.Among the major topics treated are RademacherS convergent series for the partition function.Lehner’S congruences for the Fourier coefficients of the modular functionJ，and Hecke’S theory of entire forms with multiplicative Fourier coeflicients.The last chapter gives an account of Bohr’s theory ofequivalence of general Dirichlet series.　　Both volumes of this work emphasize classical aspects of a subject which in recent years has undergone a great deaI of modern development.It iS hoped that these volumes wilI help the nonspecialist become acquainted with an important and fascinating part of mathematics and，at the same time.will provide some of the background that belongs to the repertory of every specialist in the field.　　This volume.Iike the first，iS dedicated to the students who have taken this course and have gone on to make notable contributions tO number theory and other parts of mathematics.

內容概要

　　This is the second volume of a 2-volume textbook* which evolved from a course （Mathematics 160） offered at the California Institute of Technology during the last 25 years.The second volume presupposes a background in number theory com-parable to that provided in the first volume， together with a knowledge of the basic concepts of complex analysis

書籍目錄

Chapter 1Elliptic functions  1.1   Introduction  1.2  Doubly periodic functions  1.3  Fundamental pairs of periods  1.4  Elliptic functions  1.5  Construction of elliptic functions  1.6  The Weierstrass  function  1.7  The Laurent expansion of  near the origin  1.8  Differential equation satisfied by   1.9  The Eisenstein series and the invariants  and   1.10 The numbers el, e2,e3  1.11 The discriminant   1.12 Klein's modular function J()  1.13 Invariance of J under unimodular transformations  1.14 The Fourier expansions ofg2() and g3()  1.15  The Fourier expansions of A() and J() Exercises for Chapter 1Chapter 2 The Modular group and modular functions  2.1   M6bius transformations  2.2  The modular group F　2.3  Fundamental regions　2.4  Modular functions　2.5   Special values of J　2.6   Modular functions as rational functions of J　2.7   Mapping properties of J　2.8   Application to the inversion problem for Eisenstein series　2.9   Application to Picard's theorem Exercises for Chapter 2Chapter 3　The Dedekind eta function　3.1  Introduction　3.2   Siegers proof of Theorem 3.1　3.3   Infinite product representation for A(r)　3.4   The general functional equation for q(r)　3.5   Iseki's transformation formula　3.6   Deduction of Dedekind's functional equation from Iseki's  formula　3.7   Properties of Dedekind sums　3.8   The reciprocity law for Dedekind sums　3.9   Congruence properties of Dedekind sums　3.10  The Eisenstein series G2(z)  Exercises for Chapter Chapter 4　Conyruences for the coefficients of the modular function 　4.1   Introduction　4.2   The subgroup Fo(q)　4.3   Fundamental region of Fo(p)　4.4   Functions automorphic under the subgroup Fo(p)　4.5   Construction of functions belonging to Fo(p)……Chapter 5　Rademacher's series for the partition functionChapter 6 Modular forms with multiplicative coefficientsChapter 7 Kronecker's theorem with applicationsChapter 8 General dirichlet series and Bohr's equivalence theoremSupplement to Chapter BibliographyIndex of special symbolsIndex

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