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arXiv · 1703.08827

On Dirichlet series and functional equations

Alexey Kuznetsov
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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.

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Alexey Kuznetsov. 2017-03-26. On Dirichlet series and functional equations. https://arxiv.org/abs/1703.08827

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