On the Asymptotic Behavior of the Remainder of a Dirichlet Series Absolutely Convergent in a Half-Plane

Abstract

For a Dirichlet series $\sum\nolimits_{n = 1}^\infty {a_n \exp \{ s{\lambda}_n \} }$ with nonnegative exponents and zero abscissa of absolute convergence, we study the asymptotic behavior of the remainder $\sum\nolimits_{k = n}^\infty {\left| {a_k } \right|\exp \{ {\delta \lambda}_k \} }$ , δ < 0, as n → ∞.