Dirichlet Series and L-Functions

Abstract

This chapter deals with the analytic and arithmetic properties of Dirichlet series and in particular of L-functions, of which the Riemann zeta function is the prototypical example. In a sense it is analytic number theory, but it would be inappropriate to use this expression since it now means a part of number theory that extensively uses tools from real and complex analysis, while our purpose is slightly different. Perhaps more appropriate would be “elementary number theory,” which deals with elementary number-theoretic functions, but which is also a misnomer since in no way should it be understood as “easy” number theory. In fact, the Riemann hypothesis, one of the most famous number-theoretical conjectures, can be considered as elementary number theory since it can be stated in “elementary” terms, for instance through the use of the Möbius function.