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

For a Dirichlet series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\nolimits_{n = 1}^\infty {a_n \exp \{ s{\lambda}_n \} } $$ \end{document} with nonnegative exponents and zero abscissa of absolute convergence, we study the asymptotic behavior of the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\nolimits_{k = n}^\infty {\left| {a_k } \right|\exp \{ {\delta \lambda}_k \} } $$ \end{document}, δ < 0, as n → ∞.