Analytic number theory

Includes index

Includes bibliography ( p. 599-610 ) and references.

1. Arithmetic functions
2. Elementary theory of prime numbers
3. Characters
4. Summation formulas
5. Classical analytic theory of L-functions
6. Elementary sieve methods
7. Bilinear forms and the large sieve
8. Exponential sums
9. The Dirichlet polynomials
10. Zero-density estimates
11. Sums over finite fields
12. Character sums
13. Sums over primes
14. Holomorphic modular forms
15. Spectral theory of automorphic forms
16. Sums of Kloosterman sums
17. Primes in arithmetic progressions
18. The least prime in an arithmetic progression
19. The Goldbach problem
20. The circle method
21. Equidistribution
22. Imaginary quadratic fields
23. Effective bounds for the class number
24. The critical zeros of the Riemann zeta function
25. The spacing of zeros of the Riemann zeta-function
26. Central values of L-functions

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.