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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Weighted averages of modular $L$-values
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by Andrew Knightly and Charles Li PDF
Trans. Amer. Math. Soc. 362 (2010), 1423-1443 Request permission

Abstract:

Using an explicit relative trace formula on $\operatorname {GL}(2)$, we derive a formula for averages of modular $L$-values in the critical strip, weighting by Fourier coefficients, Hecke eigenvalues, and Petersson norms. As an application we show that a GRH holds for these averages as the weight or the level goes to $\infty$. We also use the formula to give explicit zero-free regions of the form $|\operatorname {Im}(s)|\le \tau _0$ for some particular modular $L$-functions.
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Additional Information
  • Andrew Knightly
  • Affiliation: Department of Mathematics and Statistics, Neville Hall, University of Maine, Orono, Maine 04469-5752
  • Charles Li
  • Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
  • Received by editor(s): January 31, 2008
  • Published electronically: September 25, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 1423-1443
  • MSC (2000): Primary 11F11, 11F30, 11F67, 11F70
  • DOI: https://doi.org/10.1090/S0002-9947-09-04923-X
  • MathSciNet review: 2563735