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Simultaneous non-vanishing of twists


Author: Amir Akbary
Journal: Proc. Amer. Math. Soc. 134 (2006), 3143-3151
MSC (2000): Primary 11F67
Published electronically: May 18, 2006
MathSciNet review: 2231896
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Abstract: Let $ f$ be a newform of even weight $ k$, level $ M$ and character $ \psi$ and let $ g$ be a newform of even weight $ l$, level $ N$ and character $ \eta$. We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet $ L$-functions, in the context of twisted modular $ L$-functions associated to $ f$ and $ g$. Using this result, we find a lower bound in terms of $ Q$ for the number of primitive Dirichlet characters modulo prime $ q\leq Q$ whose twisted product $ L$-functions $ L_{f,\chi}(s_0) L_{g,\chi}(s_0)$ are non-vanishing at a fixed point $ s_0=\sigma_0+it_0$ with $ \frac{1}{2}<\sigma_0\leq 1$.


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  • [A] Amir Akbary, Non-vanishing of weight 𝑘 modular 𝐿-functions with large level, J. Ramanujan Math. Soc. 14 (1999), no. 1, 37–54. MR 1700874
  • [BR] Laure Barthel and Dinakar Ramakrishnan, A nonvanishing result for twists of 𝐿-functions of 𝐺𝐿(𝑛), Duke Math. J. 74 (1994), no. 3, 681–700. MR 1277950, 10.1215/S0012-7094-94-07425-5
  • [D] Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
  • [LRS] Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for 𝐺𝐿(𝑛), Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310. MR 1703764
  • [E] P. D. T. A. Elliott, On the distribution of the values of Dirichlet 𝐿-series in the half-plane 𝜎>1\over2, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 222–234. MR 0291100
  • [M] M. Ram Murty, Problems in analytic number theory, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR 1803093
  • [R] David E. Rohrlich, Nonvanishing of 𝐿-functions for 𝐺𝐿(2), Invent. Math. 97 (1989), no. 2, 381–403. MR 1001846, 10.1007/BF01389047
  • [RA1] R. A. Rankin, Sums of powers of cusp form coefficients. II, Math. Ann. 272 (1985), no. 4, 593–600. MR 807293, 10.1007/BF01455869
  • [RA2] R. A. Rankin, Contributions to the theory of Ramanujan’s function 𝜏(𝑛) and similar arithmetical functions. I. The zeros of the function ∑^{∞}_{𝑛=1}𝜏(𝑛)/𝑛^{𝑠} on the line ℜ𝔰=13/2. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351–372. MR 0000411
  • [S] Tomasz Stefanicki, Non-vanishing of 𝐿-functions attached to automorphic representations of 𝐺𝐿(2) over 𝑄, J. Reine Angew. Math. 474 (1996), 1–24. MR 1390690, 10.1515/crll.1996.474.1

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Additional Information

Amir Akbary
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive West, Lethbridge, Alberta, Canada T1K 3M4
Email: akbary@cs.uleth.ca

DOI: https://doi.org/10.1090/S0002-9939-06-08369-9
Received by editor(s): August 16, 2004
Received by editor(s) in revised form: June 9, 2005
Published electronically: May 18, 2006
Additional Notes: This research was partially supported by NSERC
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.