Author	Apostol, Tom M. author
Title	Modular Functions and Dirichlet Series in Number Theory [electronic resource] / by Tom M. Apostol
Imprint	New York, NY : Springer New York : Imprint: Springer, 1976
Connect to	http://dx.doi.org/10.1007/978-1-4684-9910-0
Descript	X, 198 p. 10 illus. online resource

This is the second volume of a 2-volume textbook* which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years. The second volume presupposes a background in number theory comยญ 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 Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular functionj( r), and Hecke's theory of entire forms with multiplicative Fourier coefficients. The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. Both volumes of this work emphasize classical aspects of a subject wh ich in recent years has undergone a great deal of modern development. It is hoped that these volumes will 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, like 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. T. M. A. January, 1976 * The first volume is in the Springer-Verlag series Undergraduate Texts in Mathematics under the title Introduction to Analytic Number Theory

1 Elliptic 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 g2 and g3 -- 1.10 The numbers el, e2, e3 -- 1.11 The discriminant ? -- 1.12 Kleinโs modular function J(?) -- 1.13 Invariance of J under unimodular transformations -- 1.14 The Fourier expansions of g2(i) and g3(i) -- 1.15 The Fourier expansions of ?(?) and J(?) -- Exercises for Chapter 1 -- 2 The Modular group and modular functions -- 2.1 Mรถbius transformations -- 2.2 The modular group ? -- 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 Picardโs theorem -- Exercises for Chapter 2 -- 3 The Dedekind eta function -- 3.1 Introduction -- 3.2 Siegelโs proof of Theorem 3.1 -- 3.3 Infinite product representation for ?(#x03C4) -- 3.4 The general functional equation for ?(?) -- 3.5 Isekiโs transformation formula -- 3.6 Deduction of Dedekindโs functional equation from Isekiโs 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(?) -- Exercises for Chapter 3 -- 4 Congruences for the coefficients of the modular function j -- 4.1 Introduction -- 4.2 The subgroup ?o(q) -- 4.3 Fundamental region of ?o(p) -- 4.4 Functions automorphic under the subgroup ?o(p) -- 4.5 Construction of functions belonging to ?o(p) -- 4.6 The behavior of fp under the generators of ? -- 4.7 The function ? (?) = ?(q?)/?(?) -- 4.8 The univalent function ?(?) -- 4.9 Invariance of ?(?) under transformations of #x03930(q) -- 4.10 The function jp expressed as a polynomial in ? -- Exercises for Chapter 4 -- 5 Rademacherโs series for the partition function -- 5.1 Introduction -- 5.2 The plan of the proof 9D -- 5.3 Dedekindโs functional equation expressed in terms of F -- 5.4 Farey fractions -- 5.5 Ford circles -- 5.6 Rademacherโs path of integration -- 5.7 Rademacherโs convergent series for p(n) -- Exercises for Chapter 5 -- 6 Modular forms with multiplicative coefficients -- 6.1 Introduction -- 6.2 Modular forms of weight k -- 6.3 The weight formula for zeros of an entire modular form -- 6.4 Representation of entire forms in terms of G4 and G6 -- 6.5 The linear space Mk and the subspace Mk,0 -- 6.6 Classification of entire forms in terms of their zeros -- 6.7 The Hecke operators Tn -- 6.8 Transformations of order n -- 6.9 Behavior of Tnf under the modular group -- 6.10 Multiplicative property of Hecke operators -- 6.11 Eigenfunctions of Hecke operators -- 6.12 Properties of simultaneous eigenforms -- 6.13 Examples of normalized simultaneous eigenforms -- 6.14 Remarks on existence of simultaneous eigenforms in M2k, 0 -- 6.15 Estimates for the Fourier coefficients of entire forms -- 6.16 Modular forms and Dirichlet series -- Exercises for Chapter 6 -- 7 Kroneckerโs theorem with applications -- 7.1 Approximating real numbers by rational numbers -- 7.2 Dirichletโs approximation theorem -- 7.3 Liouvilleโs annrnximatinn theorem -- 7.4 Kroneckerโs approximation theorem : the one-dimensional case -- 7.5 Extension of Kroneckerโs theorem to simultaneous approximation -- 7.6 Applications to the Riemann zeta function -- 7.7 Applications to periodic functions -- Exercises for Chapter 7 -- 8 General Dirichlet series and Bohrโs equivalence theorem -- 8.1 Introduction -- 8.2 The half-plane of convergence of general Dirichlet series -- 8.3 Bases for the sequence of exponents of a Dirichlet series -- 8.4 Bohr matrices -- 8.5 The Bohr function associated with a Dirichlet series -- 8.6 The set of values taken by a Dirichlet series f(s) on a line ? =?o -- 8.7 Equivalence of general Dirichlet sรฉries -- 8.8 Equivalence of ordinary Dirichlet series -- 8.9 Equality of the sets U >f(?0) and Ug(?0) for equivalent Dirichlet series -- 8.10 The set of values taken by a Dirichlet series in a neighborhood of the line ? = ?o -- 8.11 Bohrโs equivalence theorem -- 8.12 Proof of Theorem 8.15 -- 8.13 Examples of equivalent Dirichlet series. Applications of Bohrโs theorem to L-series -- 8.14 Applications of Bohrโs theorem to the Riemann zeta function -- Exercises for Chapter 8 -- Index of special symbols

1. Mathematics
2. Number theory
3. Mathematics
4. Number Theory