Abstract
In this paper we use potential theoretic arguments to establish new results concerning the overconvergence of Dirichlet series. Let \({\sum }_{j=0}^{\infty } a_{j}e^{-\lambda _{j}s}\) 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.
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I am grateful to Professor Stephen Gardiner for his valuable advice and encouragement.
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This research was supported by a Government of Ireland Postgraduate Scholarship from the Irish Research Council.
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Golitsyna, M. Overconvergence Properties of Dirichlet Series. Potential Anal 55, 1–10 (2021). https://doi.org/10.1007/s11118-020-09846-4
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DOI: https://doi.org/10.1007/s11118-020-09846-4