Analytic Number Theory

American Mathematical Soc., 2004 - 615 pages
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.

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Table des matières

Title page
Arithmetic functions
Elementary theory of prime numbers
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

Summation formulas
Classical analytic theory of Lfunctions
Elementary sieve methods
Bilinear forms and the large sieve
Exponential sums
The Dirichlet polynomials
Zerodensity estimates
Sums over finite fields
Character sums
Sums over primes
Imaginary quadratic fields
Effective bounds for the class number
The critical zeros of the Riemann zeta function
The spacing of zeros of the Riemann zetafunction
Central values of Lfunctions
Back Cover
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