This is a modern introduction to the analytic techniques used in the investigation of 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 emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.

### 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 Lindelöf 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.

### Review

‘This is a clear and concise introduction to the zeta function that concentrates on the function-theoretical aspects rather than number theory … The exercises are especially good, numerous and challenging. They extend the results of the text, or ask you to prove analogous results. Very Good Feature: Seven appendices that give most of the function-theoretical background you need to know to read this book. The Fourier Theory appendix is a gem: everything you need to know about the subject, including proofs, in 11 pages!’ Allen Stenger, Mathematical Association of America Reviews