Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Simultaneous non-vanishing of twists
HTML articles powered by AMS MathViewer

by Amir Akbary PDF
Proc. Amer. Math. Soc. 134 (2006), 3143-3151 Request permission


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 $\frac {1}{2}<\sigma _0\leq 1$.
  • Amir Akbary, Non-vanishing of weight $k$ modular $L$-functions with large level, J. Ramanujan Math. Soc. 14 (1999), no.Β 1, 37–54. MR 1700874
  • Laure Barthel and Dinakar Ramakrishnan, A nonvanishing result for twists of $L$-functions of $\textrm {GL}(n)$, Duke Math. J. 74 (1994), no.Β 3, 681–700. MR 1277950, DOI 10.1215/S0012-7094-94-07425-5
  • 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
  • Wenzhi Luo, ZeΓ©v Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for $\textrm {GL}(n)$, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp.Β 301–310. MR 1703764
  • P. D. T. A. Elliott, On the distribution of the values of Dirichlet $L$-series in the half-plane $\sigma >{1\over 2}$, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 222–234. MR 0291100
  • M. Ram Murty, Problems in analytic number theory, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR 1803093, DOI 10.1007/978-1-4757-3441-6
  • David E. Rohrlich, Nonvanishing of $L$-functions for $\textrm {GL}(2)$, Invent. Math. 97 (1989), no.Β 2, 381–403. MR 1001846, DOI 10.1007/BF01389047
  • R. A. Rankin, Sums of powers of cusp form coefficients. II, Math. Ann. 272 (1985), no.Β 4, 593–600. MR 807293, DOI 10.1007/BF01455869
  • R. A. Rankin, Contributions to the theory of Ramanujan’s function $\tau (n)$ and similar arithmetical functions. I. The zeros of the function $\sum ^\infty _{n=1}\tau (n)/n^s$ on the line ${\mathfrak {R}}s=13/2$. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), 351–372. MR 411
  • Tomasz Stefanicki, Non-vanishing of $L$-functions attached to automorphic representations of $\textrm {GL}(2)$ over $\textbf {Q}$, J. Reine Angew. Math. 474 (1996), 1–24. MR 1390690, DOI 10.1515/crll.1996.474.1
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
  • MR Author ID: 650700
  • Email:
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3143-3151
  • MSC (2000): Primary 11F67
  • DOI:
  • MathSciNet review: 2231896