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 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]**A. Akbary, Non-vanishing of weight modular -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 -functions of ,*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. 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 -series in the half plane ,*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 -functions for ,*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 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 -functions attached to automorphic representations of over ,*J. Reine Angew. Math.***474**(1996), 1-24.MR**1390690 (98a:11063)**

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.