General Dirichlet series, arithmetic convolution equations and Laplace transforms

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dc.creatorGlockner, Helge
dc.creatorLucht, Lutz G.
dc.creatorPorubsky, Stefan
dc.date2007-12-19
dc.date.accessioned2026-07-07T08:50:20Z
dc.date.available2026-07-07T08:50:20Z
dc.descriptionIn an earlier paper, we studied solutions g to convolution equations of the form a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+...+a_1*g+a_0=0, where a_0, ..., a_d are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form \sum_{x\in X} f(x) e^{-sx} (s in C^k), where X is an additive subsemigroup of [0,\infty)^k. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Feckan. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone C in R^k. Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures?
dc.description20 pages, LaTeX
dc.identifierhttps://arxiv.org/abs/0712.3172
dc.identifierhttp://arxiv.org/abs/0712.3172
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/144573
dc.subjectFunctional Analysis
dc.subjectNumber Theory
dc.subject11A25; 44A10; 46H30
dc.titleGeneral Dirichlet series, arithmetic convolution equations and Laplace transforms
dc.typetext

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