Weighted inversion of general Dirichlet series

TL;DR

This paper extends Wiener-type inversion theorems to weighted general Dirichlet series, providing new proofs and applications in analysis and number theory, including multidimensional cases.

Contribution

It introduces a weighted inversion theorem for general Dirichlet series, generalizing Edwards' 1957 result, with alternative proofs and applications.

Findings

Derived a weighted Wiener-type inversion theorem for Dirichlet series.

Provided an alternative proof using convex cone duality and extension techniques.

Discussed applications to multidimensional weighted generalized Dirichlet series.

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.