Specification of asymptotic Pólya type estimate for Dirichlet series converging in half–plane

T. I. Belous^a, A. M. Gaisin^b, R. A. Gaisin<sup>b

a</sup> Ufa University of Science and Technology, Zaki Validi str. 32, 450076, Ufa, Russia
^b Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia

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 Pólya 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 Pólya 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.

Keywords: Dirichlet series, convergence half–plane, maximal term of series, curve of bounded slope, Pólya type identity.

UDC: 517.53+517.537.72

MSC: 30D10

Received: 16.06.2024

 English version: Ufa Mathematical Journal, 2024, 16:4, 12–20