An Introduction to the Theory of the Riemann Zeta-Function

by Patterson, S. J.

• ISBN13: 9780521499057
• ISBN10: 0521499054
• Format: Paperback
• Copyright: 1995-07-01
• Publisher: Cambridge Univ Pr

Summary

This is a modern introduction to the analytic techniques used in the investigation of zeta functions, through the example of the Riemann zeta function. Riemann introduced this function in connection with his study of prime numbers and from this has developed the subject of analytic number theory. Since then many other classes of 'zeta function' have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasised central ideas of broad application, avoiding technical results and the customary function-theoretic approach. Thus, graduate students and non-specialists will find this an up-to-date and accessible introduction, especially for the purposes of algebraic number theory. There are many exercises included throughout, designed to encourage active learning.

Table of Contents

1. Historical introduction

2. The Poisson summation formula and the functional equation

3. The Hadamard product formula and 'explicit formulae' of prime number theory

4. The zeros of the zeta function and the prime number theorem

5. The Riemann hypothesis and the Lindelö

f hypothesis

6. The approximate functional equation

Appendix 1. Fourier theory

2. The Mellin transform

3. An estimate for certain integrals

4. The gamma function

5. Integral functions of finite order

6. Borel-Caratheodory theorems

7. Littlewood's theorem.