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Weighted inversion of general Dirichlet series

Authors: Helge Glöckner and Lutz G. Lucht
Journal: Trans. Amer. Math. Soc. 366 (2014), 3275-3293
MSC (2010): Primary 11M41; Secondary 30B50, 30J99, 46H99
Published electronically: November 5, 2013
MathSciNet review: 3180747
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Abstract: Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an alternative proof based on the duality theory of convex cones and extension techniques for characters of semigroups. Variants and arithmetical applications are described, including the case of multidimensional weighted generalized Dirichlet series.

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Additional Information

Helge Glöckner
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany

Lutz G. Lucht
Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstr. 1, D-38678 Clausthal-Zellerfeld, Germany

Keywords: General Dirichlet series, weighted inversion, Banach algebra, dual cone, rational vector space, separation theorem, Hahn-Banach theorem, rational polytope, semigroup algebra
Received by editor(s): December 3, 2011
Received by editor(s) in revised form: August 3, 2012, and November 13, 2012
Published electronically: November 5, 2013
Additional Notes: The first author was supported by Deutsche Forschungsgemeinschaft, GZ: GL 357/5–2.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.