This volume of the Encyclopaedia is a survey of modern number theory written by two absolutely outstanding authors. It is bound to become the standard reference in this field. Its readers will be graduate students and researchers in mathematics, logic and mathematical logic.

A unified survey of both the status quo and the continuing trends of various branches of number theory. Motivated by elementary problems, the authors present todays most significant results and methods. Topics covered include non-Abelian generalisations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. The book is rounded off with an overview of the major conjectures, most of which are based on analogies between functions and numbers, and on connections with other branches of mathematics such as analysis, representation theory, geometry and algebraic topology.

I. Problems and Tricks.- 1. Elementary Number Theory.- 2. Some Modern Problems of Elementary Number Theory.- II. Ideas and Theories.- 1. Induction and Recursion.- 2. Arithmetic of Algebraic Numbers.- 3. Arithmetic of Algebraic Varieties.- 4. Zeta Functions and Modular Forms.- References.