Analytic Number Theory

This volume presents an authoritative, up-to-date review of analytic number theory. It contains outstanding contributions from leading international figures in this field. Core topics discussed include the theory of zeta functions, spectral theory of automorphic forms, classical problems in additive number theory such as the Goldbach conjecture, and diophantine approximations and equations. This will be a valuable book for graduates and researchers working in number theory.

Foreword

vii

(2)

Speakers

ix

1. Subvarieties of Linear Tori and the Unit Equation A Survey

1

(20)

E. Bombieri

2. Remarks on the Analytic Complexity of Zeta Functions

21

(10)

E. Bombieri

3. Normal Distribution of Zeta Functions and Applications

31

(12)

E. Bombieri

A. Perelli

4. Goldbach Numbers and Uniform Distribution

43

(10)

J. Briidern

A. Perelli

5. The Number of Algebraic Numbers of Given Degree Approximating a Given Algebraic Number

53

(32)

J.H. Evertse

6. The Brun-Titchmarsh Theorem

85

(10)

J. Friedlander

H. Iwaniec

7. A Decomposition of Riemann's Zeta-Function

95

(8)

A. Granville

8. Multiplicative Properties of Consecutive Integers

103

(16)

A.J. Hildebrand

9. On the Equation (x(m)-1)/(x-1) = y(q) with x Power

119

(8)

N. Hirata-Kohno

T.N. Shorey

10. Congruence Families of Exponential Sums

127

(12)

M.N. Huxley

N. Watt

11. On Some Results Concerning the Riemann Hypothesis

139

(30)

A. Ivic

12. Mean Values of Dirichlet Series via Laplace Transforms

169

(40)

M. Jutila

13. The Mean Square of the Error Term in a Genelarization of the Dirichlet Divisor Problem

209

(18)

K.-Y. Lam

K.-M. Tsang

14. The Goldbach Problem with Primes in Arithmetic Progressions

227

(26)

M.-C. Liu

T. Zhan

15. On the Sum of Three Squares of Primes

253

(12)

H. Mikawa

16. Trace Formula over the Hyperbolic Upper Half Space

265

(22)

Y. Motohashi

17. Modular Forms and the Chebotarev Density Theorem II

287

(22)

V.K. Murty

18. Congruences between Modular Forms

309

(12)

M.R. Murty

19. Regular Singularities in G-Function Theory

321

(16)

M. Nagata

20. Spectral Theory and L-functions

337

(6)

P. Sarnak

21. Irrationality Criteria for Numbers of Mahler's Type

343

(10)

T.N. Shorey

R. Tijdeman

22. Hypergeometric Functions and Irrationality Measures

353

(8)

C. Viola

23. Forms in Many Variables

361

(16)

T.D. Wooley

24. Remark on the Kuznetsov Trace Formula

377

E. Yoshida

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Analytic Number Theory

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Table of Contents

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