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

HTML articles powered by AMS MathViewer

- by J. Kaczorowski and A. Perelli PDF
- Proc. Amer. Math. Soc.
**130**(2002), 2821-2826 Request permission

## 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 $\mathcal {S}^\sharp$.## References

- V. Bernstein, Séries de Dirichlet, Gauthier-Villars 1933.
- J. B. Conrey and A. Ghosh,
*On the Selberg class of Dirichlet series: small degrees*, Duke Math. J.**72**(1993), no. 3, 673–693. MR**1253620**, DOI 10.1215/S0012-7094-93-07225-0 - 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. - Jerzy Kaczorowski and Alberto Perelli,
*On the structure of the Selberg class. I. $0\leq d\leq 1$*, Acta Math.**182**(1999), no. 2, 207–241. MR**1710182**, DOI 10.1007/BF02392574 - J. Kaczorowski and A. Perelli,
*The Selberg class: a survey*, Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 953–992. MR**1689554** - J. Kaczorowski and A. Perelli,
*On the structure of the Selberg class. III. Sarnak’s rigidity conjecture*, Duke Math. J.**101**(2000), no. 3, 529–554. MR**1740688**, DOI 10.1215/S0012-7094-00-10137-8 - Hidegorô Nakano,
*Über Abelsche Ringe von Projektionsoperatoren*, Proc. Phys.-Math. Soc. Japan (3)**21**(1939), 357–375 (German). MR**94** - Atle Selberg,
*Old and new conjectures and results about a class of Dirichlet series*, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno, 1992, pp. 367–385. MR**1220477** - Ulrike M. A. Vorhauer and Eduard Wirsing,
*On Sarnak’s rigidity conjecture*, J. Reine Angew. Math.**531**(2001), 35–47. MR**1810115**, DOI 10.1515/crll.2001.008

## Additional Information

**J. Kaczorowski**- Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, 60-769 Poznań, Poland
- MR Author ID: 96610
- Email: kjerzy@math.amu.edu.pl
**A. Perelli**- Affiliation: Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy
- MR Author ID: 137910
- Email: perelli@dima.unige.it
- Received by editor(s): May 11, 2001
- Published electronically: April 17, 2002
- Communicated by: Dennis A. Hejhal
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**130**(2002), 2821-2826 - MSC (2000): Primary 11M41
- DOI: https://doi.org/10.1090/S0002-9939-02-06542-5
- MathSciNet review: 1908263