Specification of asymptotic Polya type estimate for Dirichlet series converging in half-plane

Abstract

We study the asymptotic behavior of a Dirichlet series with positive exponents, converging in the left half--plane, on an arc of bounded slope ending on the convergence line. In the paper we obtain conditions under which the sum of the Dirichlet series satisfies an asymptotic equality of Polya type on a set, the upper density of which is equal to one.
In 2023 we obtained results related to dual cases. We showed that a Polya type identity holds on an asymptotic set of positive upper density depending on the slope coefficient (Lipschitz constant) of the arc. In this paper, we prove a common theorem covering both of these cases, and we show that the asymptotic set has an upper density, which is equal to one.