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Applications of a theorem of H. Cramér to the Selberg class


Authors: J. Kaczorowski and A. Perelli
Journal: Proc. Amer. Math. Soc. 130 (2002), 2821-2826
MSC (2000): Primary 11M41
DOI: https://doi.org/10.1090/S0002-9939-02-06542-5
Published electronically: April 17, 2002
MathSciNet review: 1908263
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Abstract: We prove two results on the nature of the Dirichlet coefficients $a(n)$ of the $L$-functions in the extended Selberg class $\mathcal{S}^\sharp$. The first result asserts that if $a(n)=\phi(\log n)$ for some entire function $\phi(z)$of order 1 and finite type, then $\phi(z)$ is constant. The second result states, roughly, that if $a(n)\phi(\log n)$ are still the coefficients of some $L$-function from $\mathcal{S}^\sharp$, then $\phi(z)=ce^{i\beta z}$ with $c\in\mathbb{C}$ and $\beta\in\mathbb{R}$. The proofs are based on an old result by Cramér and on the characterization of the functions of degree 1 of $\mathcal{S}^\sharp$.


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Additional Information

J. Kaczorowski
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, 60-769 Poznań, Poland
Email: kjerzy@math.amu.edu.pl

A. Perelli
Affiliation: Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy
Email: perelli@dima.unige.it

DOI: https://doi.org/10.1090/S0002-9939-02-06542-5
Received by editor(s): May 11, 2001
Published electronically: April 17, 2002
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2002 American Mathematical Society

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