A well known theorem by Hamburger, see cite{Ham1} and Ch. II of Titchmarsh \cite{Tit1}, states that the Riemann zeta function $\zeta(s)$ is determined by its functional equation in the following sense. Let $f(s)$ and $g(s)$ be two Dirichlet series absolutely convergent for $\sigma>1$ such that $(s-1)f(s)$ and $(s-1)g(s)$ are entire functions of finite order, and $f(s)$ and $g(s)$ satisfy the functional equation $\pi^{-s/2}\Gamma\big(\frac{s}{2}\big) f(s) = \pi^{-(1-s)/2} \Gamma\big(\frac{1-s}{2}\big) g(1-s).$ Then $f(s) = g(s) =c\zeta(s)$ for some $c\in\CC$. In fact, the same conclusion holds under weaker conditions on $f(s)$ and $g(s)$; we refer to Piatetski-Shapiro and Raghunathan \cite{PS-R1}, and to the literature quoted there, for an interesting discussion of the above theorem, especially in connection with uniqueness properties of the Poisson summation formula.

A converse theorem for Dirichlet L-functions / J. Kaczorowski, G. Molteni, A. Perelli. - In: COMMENTARII MATHEMATICI HELVETICI. - ISSN 0010-2571. - 85:2(2010), pp. 463-483. [10.4171/CMH/202]

### A converse theorem for Dirichlet L-functions

#### Abstract

A well known theorem by Hamburger, see cite{Ham1} and Ch. II of Titchmarsh \cite{Tit1}, states that the Riemann zeta function $\zeta(s)$ is determined by its functional equation in the following sense. Let $f(s)$ and $g(s)$ be two Dirichlet series absolutely convergent for $\sigma>1$ such that $(s-1)f(s)$ and $(s-1)g(s)$ are entire functions of finite order, and $f(s)$ and $g(s)$ satisfy the functional equation $\pi^{-s/2}\Gamma\big(\frac{s}{2}\big) f(s) = \pi^{-(1-s)/2} \Gamma\big(\frac{1-s}{2}\big) g(1-s).$ Then $f(s) = g(s) =c\zeta(s)$ for some $c\in\CC$. In fact, the same conclusion holds under weaker conditions on $f(s)$ and $g(s)$; we refer to Piatetski-Shapiro and Raghunathan \cite{PS-R1}, and to the literature quoted there, for an interesting discussion of the above theorem, especially in connection with uniqueness properties of the Poisson summation formula.
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Dirichlet L-functions ; converse theorems ; Hamburger theorem ; Selberg class
Settore MAT/05 - Analisi Matematica
COMMENTARII MATHEMATICI HELVETICI
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/142161
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