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Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)?

I am aware of the coefficient forumla but it does not seem to be practical for elementary functions.

It seems that the Dirichlet series are specific to the number theory and completely avoid the domain of mathematical analysis, with the exception of the Zeta function (which is however not elementary).

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Dirichlet series do pop up in analysis, specifically in basis theory: Hedenmalm, Lindqvist and Seip used them to solve the "dilation basis problem" in $L^2[0,1]$ in this paper: https://arxiv.org/abs/math/9512211.

To answer your first question: you can rewrite any function f(z) that is analytic on B(0,1), i.e. the open disk with radius 1, as a Dirichlet series g(s) defined on the open half-plane $Rs > 0$ through the change of variable $z = p^{-s}$, for some natural number $p$.

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  • $\begingroup$ It might be very helpful (for such hobby mathematicians as me) to provide a worked example (for e.g. $y=x$ or $y=\exp(x)$). I do not understand the substitution. Should it be applied to the power series? It gives $\sum_n a_n z^n = \sum_n a_n/p^{-sn}$. The latter does not look like a Dirichlet series, where the running index is in the base (and not the exponent) of the expression. $\endgroup$
    – F. Jatpil
    Commented Dec 18, 2023 at 8:51
  • $\begingroup$ Dirichlet series are of the form $\sum_n a_nn^{-s}$. For a general powerseries $\sum_n a_n z^n$ the substitution yields $\sum _n a_n p^{-ns}$. This is a Dirichlet series: choose coefficients $b_k = 0$ if $k$ is not of the form $p^n$ and let $b_{p^n} = a_n$. Then $\sum_n a_n p^{-ns} = \sum_k b_k k^{-s}$. $\endgroup$
    – Domenic Petzinna
    Commented Dec 19, 2023 at 8:40

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