Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [A] A. Akbary, Non-vanishing of weight $ k$ modular $ L$-functions with large level, J. Ramanujan Math. Soc. 14 (1999), 37-54. MR 1700874 (2000e:11067)
  • [BR] L. Barthel and D. Ramakrishnan, A nonvanishing result for twists of $ L$-functions of $ {\text GL}(n)$, Duke Math. J. 74 (1994), 681-700.MR 1277950 (95d:11062)
  • [D] H. Davenport, Multiplicative Number Theory, third edition, Springer, 2000.MR 1790423 (2001f:11001)
  • [LRS] W. Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for GL$ (n)$. In Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), 301-310, Proc. Sympos. Pure Math., 66, Part 2, AMS, 1999.MR 1703764 (2000e:11072)
  • [E] P. D. T. A. Elliott, On the distribution of the values of Dirichlet $ L$-series in the half plane $ \sigma>\frac{1}{2}$, Indag. Math. 33 (1971), 222-234.MR 0291100 (45:194)
  • [M] M. R. Murty, Problems in Analytic Number Theory, Springer, 2001. MR 1803093 (2001k:11002)
  • [R] D. Rohrlich, Nonvanishing of $ L$-functions for $ {\text GL}(2)$, Invent. Math. 97 (1989), 381-403. MR 1001846 (90g:11062)
  • [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 $ \tau(n)$ and similar arithmetical functions. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Phil. Soc. 35 (1939), 357-372.MR 0000411 (1:69d)
  • [S] T. 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 (98a:11063)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F67

Retrieve articles in all journals with MSC (2000): 11F67


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.

American Mathematical Society