On Dirichlet series and functional equations

Alexey Kuznetsov

Published 2017-03-26Version 1

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solutions of certain functional equations. Our result seems to be a Dirichlet series analogue of the well known Lagrange-Burmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of Levy processes.