Overconvergence Properties of Dirichlet Series

In this paper we use potential theoretic arguments to establish new results concerning the overconvergence of Dirichlet series. Let ∑j=0∞aje−λjs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\sum }_{j=0}^{\infty } a_{j}e^{-\lambda _{j}s}$\end{document} converge on the half-plane {Re(s) > 0} to a holomorphic function f. Our first result gives sufficient conditions for a subsequence of partial sums of the series to converge at every regular point of f. The second result shows, in particular, that if a subsequence of the partial sums of the series is uniformly bounded on a nonpolar compact set K ⊂{Re(s) < 0} and ξ ∈{Re(s) = 0} is a regular point of f, then this subsequence converges on a neighbourhood of ξ.