On Dirichlet series and functional equations

TL;DR

This paper introduces a novel Dirichlet series representation for solutions to a specific functional equation involving L-functions, employing probabilistic methods related to Lévy processes, expanding the analytical tools for Dirichlet series.

Contribution

It provides a new Dirichlet series analogue of the Lagrange-Bürmann formula for a class of functional equations involving L-functions, using probabilistic techniques.

Findings

Derived Dirichlet series representation for solutions to the functional equation

Connected the representation to a probabilistic analogue of the Lagrange-Bürmann formula

Utilized Kendall's identity from Lévy process fluctuation theory

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-B\"urmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of L\'evy processes.