Simultaneous non-vanishing of twists

Author:
Amir Akbary

Journal:
Proc. Amer. Math. Soc. **134** (2006), 3143-3151

MSC (2000):
Primary 11F67

DOI:
https://doi.org/10.1090/S0002-9939-06-08369-9

Published electronically:
May 18, 2006

MathSciNet review:
2231896

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Abstract: Let be a newform of even weight , level and character and let be a newform of even weight , level and character . We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet -functions, in the context of twisted modular -functions associated to and . Using this result, we find a lower bound in terms of for the number of primitive Dirichlet characters modulo prime whose twisted product -functions are non-vanishing at a fixed point with .

**[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**, https://doi.org/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**, https://doi.org/10.1007/BF01389047**[RA1]**R. A. Rankin, Sums of powers of cusp form coefficients, II,*Math. Ann.***272**(1985), 593-600. MR**0807293 (87d:11032)****[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**, https://doi.org/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.