Journal
AMERICAN JOURNAL OF MATHEMATICS
Volume 139, Issue 3, Pages 707-768Publisher
JOHNS HOPKINS UNIV PRESS
DOI: 10.1353/ajm.2017.0019
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Funding
- ERC [258713, 228304]
- Volkswagen Foundation
- ETH Zurich
- EPF Lausanne
- Institut Universitaire de France
- SNF [200021-137488]
- DFG-SNF lead agency program [200021L_153647]
- NSA [H98230-14-1-0139]
- NSF [DMS-1503629]
- ARC [DP130100674]
- Swiss National Science Foundation (SNF) [200021L_153647] Funding Source: Swiss National Science Foundation (SNF)
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1503629] Funding Source: National Science Foundation
- European Research Council (ERC) [258713] Funding Source: European Research Council (ERC)
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We study the average of the product of the central values of two L-functions of modular forms f and g twisted by Dirichlet characters to a large prime modulus q. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q, and we use the theory of Deligne and Katz to prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of q. When at least one of the forms f and g is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted L-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M. Young's asymptotic evaluation of the fourth moment of Dirichlet L-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.
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