General Dirichlet series, arithmetic convolution equations and Laplace transforms
Volume 193 / 2009
Abstract
In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions $g\colon\mathbb N\to\mathbb C$ to convolution equations of the form \[ a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+\cdots+a_1*g+a_0=0, \] where $a_0,\ldots,a_d\colon\mathbb N\to\mathbb C$ 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 $\sum_{x\in X}f(x)e^{-sx}$ ($s\in \mathbb C^k$), where $X\subseteq [0,\infty)^k$ is an additive subsemigroup. If $X$ is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Fečkan [Proc. Amer. Math. Soc. 136 (2008)]. 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 $C\subseteq \mathbb R^k$. 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?
