On Dirichlet series and functional equations

2017/03/26 by Alexey Kuznetsov, Kuznetsov, Alexey
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 11M41 #Secondary 60G51

paper · pdf · doi:10.48550/arxiv.1703.08827

openalex publication_date 2017/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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