An introduction to the theory of the Riemann zeta-function.

*(English)*Zbl 0641.10029
Cambridge Studies in Advanced Mathematics, 14. Cambridge etc.: Cambridge University Press. XIII, 156 p. (1988).

The theory of the Riemann zeta-function and its generalizations represent one of the most beautiful developments in number theory. It is the connection between the continuous and the discrete that is so attractive.

As the author points out the aims of this book are rather different to other books, such as the classics of Landau, Titchmarsh, Edwards or Ivić. It is rather to bring out the central role of the Poisson summation formula and the ‘explicit formulae’ of prime number theory. Therefore the ‘Riemann hypothesis’ is discussed at much greater length than is usual, just in the spirit of Weil.

The main part of this book consists of 6 chapters: (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 (with an excellent written introduction to the theory of zeta-functions attached to curves over finite fields), (6) The approximate functional equation.

Seven appendices contain some information which is usually not studied in undergraduate courses. In general the reader with a good background in analysis and number theory is able to read this book without having to refer to the appendices too often. Very valuable is the decision of the author to insert a large number of exercises, of very different levels of difficulty.

The book is carefully printed, only a few misprints seem to exist, for example the definition of F(x) on page 25.

As the author points out the aims of this book are rather different to other books, such as the classics of Landau, Titchmarsh, Edwards or Ivić. It is rather to bring out the central role of the Poisson summation formula and the ‘explicit formulae’ of prime number theory. Therefore the ‘Riemann hypothesis’ is discussed at much greater length than is usual, just in the spirit of Weil.

The main part of this book consists of 6 chapters: (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 (with an excellent written introduction to the theory of zeta-functions attached to curves over finite fields), (6) The approximate functional equation.

Seven appendices contain some information which is usually not studied in undergraduate courses. In general the reader with a good background in analysis and number theory is able to read this book without having to refer to the appendices too often. Very valuable is the decision of the author to insert a large number of exercises, of very different levels of difficulty.

The book is carefully printed, only a few misprints seem to exist, for example the definition of F(x) on page 25.

Reviewer: H.Müller

##### MSC:

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14G15 | Finite ground fields in algebraic geometry |