The theory of the Riemann zeta-function and its generalisations represent one of the most beautiful developments in mathematics. The Riemann zeta-function is a meromorphic function whose properties can on the one hand be investigated by the techniques of complex analysis, and on the other yield difficult theorems concerning the integers. It is this connection between the continuous and the discrete that is so wonderful. It is the purpose of this book to explain this connection through the example of the Riemann zeta-function. The more general zeta- and L-functions will not be introduced but the reader who has studied the techniques described here should have no trouble in seeing how they apply in a more general context. This book is intended as an introduction; there are developments in so many directions that could have been followed, but which, in the interest of conciseness, have not.

The Riemann zeta-function belongs to ‘classical’ mathematics, and the development of the theory here is essentially classical. Naturally this is not the first book on this subject, nor will it be the last. The aims of this book are rather different to other books, such as the classics of Landau (Landau (1)) and Titchmarsh (Titchmarsh (2)) to which the reader interested in the finer theory will turn sooner or later, or the historical treatise of Edwards (Edwards (1)), or Ivić's book (Ivić (1)) on the most delicate modern results.