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On the Non-Vanishing of a Certain Class of Dirichlet Series

Published online by Cambridge University Press:  20 November 2018

Sridhar Narayanan
Sridhar Narayanan*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6
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Abstract

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In this paper, we consider Dirichlet series with Euler products of the form F(s) = Πp in > 1, and which are regular in ≥ 1 except for a pole of order m at s = 1. We establish criteria for such a Dirichlet series to be nonvanishing on the line of convergence. We also show that our results can be applied to yield non-vanishing results for a subclass of the Selberg class and the Sato-Tate conjecture.

Keywords

Primary: 11MxxSecondary: 11M41
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 3 , 01 September 1997 , pp. 364 - 369
Copyright
Copyright © Canadian Mathematical Society 1997

References

[C-G] Conrey, B. and Ghosh, A., Selberg class of Dirichlet series: Small degrees, Duke Math. J. 72 (1993), 673–693.Google Scholar
[K] Knapp, A., Elliptic Curves, Princeton University Press, Princeton, 1992.Google Scholar
[KM] Kumar Murty, V., On the Sato-Tate conjecture. In: Number Theory related to Fermat's last theorem, (ed. N. Koblitz), 1982, 195–205.Google Scholar
[R] Rankin, R. A., Contributions to the theory of Ramanujan's function ú(n) and similar arithmetical functions I and II, Proceedings of Cambridge Philosophical Society 35 (1939), 351–372.Google Scholar
[RM] Ram Murty, M., Selberg's conjectures and Artin L-functions, Bull. Amer.Math. Soc. (N.S.) (1) 31 (1994).Google Scholar
[S] Serre, J. P., Abelian l-adic representations and Elliptic curves, McGill University lecture notes, (written in collaboration with W. Kuyk and J. Labute), W. A. Benjamin Inc., New York, Amsterdam, 1968.Google Scholar
[SG] Selberg, A., Old and new conjectures and results about a certain class of Dirichlet series, Collected Papers, II, 47–63.Google Scholar