On mean values and zeroes of Dirichlet series

TL;DR

This paper investigates the mean values and zeroes of a broad class of Dirichlet series, including the Riemann zeta-function and L-functions, introducing a new perspective on their mean value half-plane and confirming an analog of Lindelöf's Hypothesis.

Contribution

It introduces a new approach to defining the half-plane of mean values for Dirichlet series and proves that regular series in this class have mean values, supporting an analog of Lindelöf's Hypothesis.

Findings

In the half-plane of mean values, regular series have mean values.

The analog of Lindelöf's Hypothesis is confirmed.

The half-plane free of zeroes is identified for series with inverse functions.

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.