On mean values and zeroes of Dirichlet series

Abstract

In this paper we study the mean values and zeroes of Dirichlet series of a view $\sum_{n}a_n n^{-s}$ with complex coefficients. There was introduced some class of Dirichlet series including such widely used series as the Riemann zeta-function, Dirichlet L-functions and ets. A new point of view is introduced in defining of a half plane of mean values. It was proven that in the half plane of mean values any natural degree of the series of an inroduced class, being regular in this half plane,has a mean value. In particular, the analog of Lindel\"of Hypothesis is true. If, in addition, the Dirichlet series f(s) belongs to this class with the function f(s)^{-1} then the half plane of mean values was proved to be free from the zeroes.