On Dirichlet series and functional equations

Abstract

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 solution f=f(s,w) of the functional equation L(s−wf)=exp⁡(f), where L(s) is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange–Bürmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of Lévy processes.