Analytic Number Theory distinguishes itself by the variety of tools it uses to establish results. One of the primary attractions of this theory is its vast diversity of concepts and methods. The main goals of this book are to show the scope of the theory, both in classical and modern directions, and to exhibit its wealth and prospects, beautiful theorems, and powerful techniques.

The book is written with graduate students in mind, and the authors nicely balance clarity, completeness, and generality. The exercises in each section serve dual purposes, some intended to improve readers' understanding of the subject and others providing additional information. Formal prerequisites for the major part of the book do not go beyond calculus, complex analysis, integration, and Fourier series and integrals. In later chapters automorphic forms become important, with much of the necessary information about them included in two survey chapters.

Readership

Graduate students and research mathematicians interested in analytic number theory.

Reviews

"The book is written in a very lively and nicely readable style ... contains a very well chosen and balanced material."

-- EMS Newsletter

"The authors are active researchers with a lot of experience and deep insight, and their creative attitude makes reading particularly rewarding... It can be warmly recommended toa wide readership ..."

-- Zentralblatt Math

Table of Contents

• Introduction
• Arithmetic functions
• Elementary theory of prime numbers
• Characters
• Summation formulas
• Classical analytic theory of $L$-functions
• Elementary sieve methods
• Bilinear forms and the large sieve
• Exponential sums
• The Dirichlet polynomials
• Zero-density estimates
• Sums over finite fields
• Character sums
• Sums over primes
• Holomorphic modular forms
• Spectral theory of automorphic forms
• Sums of Kloosterman sums
• Primes in arithmetic progressions
• The least prime in an arithmetic progression
• The Goldbach problem
• The circle method
• Equidistribution
• Imaginary quadratic fields
• Effective bounds for the class number
• The critical zeros of the Riemann zeta function
• The spacing of zeros of the Riemann zeta-function
• Central values of $L$-functions
• Bibliography
• Index