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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
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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  • [1] V. Bernstein, Séries de Dirichlet, Gauthier-Villars 1933.
  • [2] J. B. Conrey, A. Ghosh, On the Selberg class of Dirichlet series: small degrees, Duke Math. J. 72 (1993), 673-693. MR 95f:11064
  • [3] H. Cramér, Un théorème sur les séries de Dirichlet et son application, Ark. Mat. Astr. Fys. 13 (1918), 1-14; Collected Works, vol I, 71-84, Springer Verlag 1994.
  • [4] J. Kaczorowski, A. Perelli, On the structure of the Selberg class, I: $0\leq d\leq 1$ - Acta Math. 182 (1999), 207-241. MR 2000h:11097
  • [5] J. Kaczorowski, A. Perelli, The Selberg class: a survey, Number Theory in Progress, Proc. Conf. in Honor of A. Schinzel, ed. by K. Györy et al., 953-992, de Gruyter 1999. MR 2001g:11141
  • [6] J. Kaczorowski, A. Perelli, On the structure of the Selberg class, III: Sarnak's rigidity conjecture, Duke Math. J. 101 (2000), 529-554. MR 2001g:11140
  • [7] M. R. Murty, Selberg's conjectures and Artin $L$-functions, Bull. A. M. S. 31 (1994), 1-14. MR 94j:11116
  • [8] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. Amalfi Conf. Analytic Number Theory, ed. by E. Bombieri et al., 367-385, Università di Salerno 1992; Collected Papers, vol. II, 47-63, Springer Verlag 1991. MR 94f:11085
  • [9] U. M. A. Vorhauer, E. Wirsing On Sarnak's rigidity conjecture, J. reine angew. Math. 531 (2001), 35-47. MR 2001k:11176

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Additional Information

J. Kaczorowski
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, 60-769 Poznań, Poland

A. Perelli
Affiliation: Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy

Received by editor(s): May 11, 2001
Published electronically: April 17, 2002
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2002 American Mathematical Society