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Transactions of the American Mathematical Society

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Weighted averages of modular $ L$-values

Authors: Andrew Knightly and Charles Li
Journal: Trans. Amer. Math. Soc. 362 (2010), 1423-1443
MSC (2000): Primary 11F11, 11F30, 11F67, 11F70
Published electronically: September 25, 2009
MathSciNet review: 2563735
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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 $ \vert\operatorname{Im}(s)\vert\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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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