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arXiv · 1905.04193

A uniqueness property of general Dirichlet series

Anup B. Dixit
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Abstract

Let $F(s)=\sum_n a_n/λ_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $α_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $α_F$ and residue $ρ_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$.

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Anup B. Dixit. 2019-05-10. A uniqueness property of general Dirichlet series. https://doi.org/10.1016/j.jnt.2019.06.007

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