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 \(F(s) = \sum\limits_n {\kern 1pt} {{a}_{n}}{{e}^{{{{\lambda }_{n}}s}}},\) \(0 < {{\lambda }_{n}} \uparrow \infty ,\) 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 the type of Pólya for the sum F(s) of the Dirichlet series be valid in the case when the argument \(s\) tends to the imaginary axis along \(\gamma \) over a sufficiently massive set.