Polya Type Equality for Dirichlet Series of the Class \(\boldsymbol{D}\)(\(\boldsymbol{\Phi}\)) on Curves

We study the asymptotic behavior of the sum of entire Dirichlet series with positive exponents on curves of a bounded slope going in a certain way to infinity. For entire transcendental functions of finite order Polia showed that if the density of the sequence of exponents is equal to zero, then for any curve going to infinity, there is an unlimited sequence of points on which the logarithm of the modulus of the sum of the series is equivalent to the logarithm of the maximum of the modulus. Later, these results were completely transferred by I.D. Latypov to entire Dirichlet series of finite order and finite lower order by Ritt. A further generalization was obtained in the works of N.N. Yusupova-Aitkuzhina to the more general dual classes of Dirichlet series defined by the convex majorant. In this paper, we obtained necessary and sufficient conditions for the exponents under which the logarithm of the modulus of the sum of any Dirichlet series from one such class on a curve of bounded slope is equivalent to the logarithm of the maximum term on an asymptotic set whose upper density is equal to one.