An estimate for the sum of a Dirichlet series on an arc of bounded slope

Abstract

The article considers the behavior of the sum of the Dirichlet series F(s) = \sum nane\lambda ns, 0 < \lambda n \uparrow \infty , which converges absolutely in the left half-plane \Pi 0, on a curve arbitrarily approaching the imaginary axis — the boundary of this half-plane. We have obtained a solution to the following problem: Under what additional conditions on \gamma will the strengthened asymptotic relation be valid in the case when the argument s tends to the imaginary axis along \gamma over a sufficiently massive set.