General Dirichlet series, arithmetic convolution equations and Laplace transforms

TL;DR

This paper extends the study of convolution equations involving Dirichlet series to multidimensional cases, exploring solutions via recursive methods and connecting to Radon measures and Laplace transforms in convex cones.

Contribution

It generalizes previous results to multidimensional Dirichlet series and introduces a recursive solution approach for discrete semigroups, also linking to Radon measures and Laplace transforms.

Findings

Solutions can be determined recursively for discrete semigroups.

The work connects convolution equations to Radon measures and Laplace transforms.

Provides a framework for solving polynomial equations involving Laplace transforms in convex cones.

Abstract

In 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?