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1990 | Buch | 2. Auflage

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Modular Functions and Dirichlet Series in Number Theory

verfasst von: Tom M. Apostol

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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. 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 which 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.

Inhaltsverzeichnis

Frontmatter

1. Elliptic functions

Abstract

Additive number theory is concerned with expressing an integer n as a sum of integers from some given set S. For example, S might consist of primes, squares, cubes, or other special numbers. We ask whether or not a given number can be expressed as a sum of elements of S and, if so, in how many ways this can be done.

Tom M. Apostol

2. The modular group and modular functions

Abstract

In the foregoing chapter we encountered unimodular transformations

$$ {{c\tau + d}} $$

where a, b, c, d are integers with ad — bc = 1. This chapter studies such transformations in greater detail and also studies functions which, Iike J(τ), are invariant under unimodular transformations. We begin with some remarks concerning the more general transformations

$$ {{cz + d}} $$

(1)

where a, b, c, d are arbitrary complex numbers.

Tom M. Apostol

3. The Dedekind eta function

Abstract

In many applications of elliptic modular functions to number theory the eta function plays a central role. It was introduced by Dedekind in 1877 and is defined in the half-plane H = {τ: Im(τ) > 0} by the equation

$$ \eta (\tau ) = e^{\pi i\tau /12} \prod\limits_{n = 1}^\infty {\left( {1 - e2^{\pi in\tau } } \right)} . $$

(1)

The infinite product has the form Π (1 — xⁿ) where x = e^2πiτ. If τ∈H then |x| < 1 so the product converges absolutely and is nonzero. Moreover, since the convergence is uniform on compact subsets of H, η(τ) is analytic on H.

Tom M. Apostol

4. Congruences for the coefficients of the modular function j

Abstract

The functionj(τ) = 12³ J(τ) has a Fourier expansion of the form

$$ {x} + \sum\limits_{n = \left. 0 \right|}^\infty {c(n)x^n ,\,(x = e^{2\pi i\tau } )} $$

where the coefficients c(n) are integers. At the end of Chapter 1 we mentioned a number of congruences involving these integers. This chapter shows how so me of these congruences are obtained. Specifically we will prove that

$$ \begin{array}{*{20}c} {c(2n) \equiv 0(\bmod 2^{11} ),} \\ {c(3n) \equiv 0(\bmod 3^5 ),} \\ {c(5n) \equiv 0(\bmod 5^2 ),} \\ {c(7n) \equiv 0(\bmod 7).} \\ \end{array} $$

The method used to obtain these congruences can be illustrated for the modulus 5². We consider the function

$$ f_5 (\tau ) = \sum\limits_{n = 1}^\infty {c(5n)x^n } $$

obtained by extracting every fifth coefficient in the Fourier expansion of j. Then we show that there is an identity of the form

$$ f_5 (\tau ) = 25\left\{ {a_1 \Phi (\tau ) + a_2 \Phi ^2 (\tau ) + \cdots + a_k \Phi ^k (\tau )} \right\}, $$

(1)

where the a_i are integers and Ф(τ) has a power series expansion in x = e^2πiτ with integer coefficients. By equating coefficients in (1) we see that each coefficient of f₅(τ) is divisible by 25.

Tom M. Apostol

5. Rademacher’s series for the partition function

Abstract

The unrestricted partition function p(n) counts the number of ways a positive integer n can be expressed as a sum of positive integers ≤n. The number of summands is unrestricted, repetition is allowed, and the order of the summands is not taken into account.

Tom M. Apostol

6. Modular forms with multiplicative coefficients

Abstract

The material in this chapter is motivated by properties shared by the discriminant Δ(τ) and the Eisenstein series

$$ {G_{2k}}(\tau ) = \sum\limits_{(m,n) \ne (0,0)} {\frac{1}{{{{(m + n\tau )}^{2k,}}}}} $$

where k is an integer, k ≥ 2. All these functions satisfy the relation

$$ f(\frac{{a\tau + b}}{{c\tau + b}}) = {(c\tau + d)^r}f(\tau ), $$

(1)

where r is an integer and $ (\begin{array}{*{20}{c}} a&b \\ c&d \end{array}) $ is any element of the modular group Γ.

Tom M. Apostol

7. Kronecker’s theorem with applications

Abstract

Every irrational number θ can be approximated to any desired degree of accuracy by rational numbers. In fact, if we truncate the decimal expansion of θ after n decimal places we obtain a rational number which differs from θ by less than 10^-n. However, the truncated decimals might have very large denominators. For example, if

$$ \theta = \pi - 3 = 0.141592653 \ldots $$

the first five decimal approximations are 0.1, 0.14, 0.141, 0.1415, 0.14159. Written in the form a/b, where a and b are relatively prime integers, these rational approximations become

$$ \frac{1} {{10}},\frac{7} {{50}},\frac{{141}} {{1000}},\frac{{283}} {{100,000}}. $$

On the other hand, the fraction 1/7 = 0.142857 … differs from θ by less than 2/1000 and is nearly as good as 141/1000 for approximating θ, yet its denominator 7 is very small compared to 1000.

Tom M. Apostol

8. General Dirichlet series and Bohr’s equivalence theorem

Abstract

This chapter treats a class of series, called general Dirichlet series, which includes both power series and ordinary Dirichlet series as special cases. Most of the chapter is devoted to a method developed by Harald Bohr [6] in 1919 for studying the set of values taken by Dirichlet series in a half-plane. Bohr introduced an equivalence relation among Dirichlet series and showed that equivalent Dirichlet series take the same set of values in certain half-planes. The theory uses Kronecker’s approximation theorem discussed in the previous chapter. At the end of the chapter applications are given to the Riemann zeta function and to Dirichlet L-functions.

Tom M. Apostol

Backmatter

Titel: Modular Functions and Dirichlet Series in Number Theory
verfasst von: Tom M. Apostol
Verlag: Springer New York
Electronic ISBN: 978-1-4612-0999-7
Print ISBN: 978-1-4612-6978-6
DOI: https://doi.org/10.1007/978-1-4612-0999-7

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Inhaltsverzeichnis

Frontmatter

1. Elliptic functions

2. The modular group and modular functions

3. The Dedekind eta function

4. Congruences for the coefficients of the modular function j

5. Rademacher’s series for the partition function

6. Modular forms with multiplicative coefficients

7. Kronecker’s theorem with applications

8. General Dirichlet series and Bohr’s equivalence theorem

Backmatter