Petrow, I;
(2013)
Transition mean values of shifted convolution sums.
Journal of Number Theory
, 133
(10)
pp. 3264-3282.
10.1016/j.jnt.2013.04.003.
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Abstract
Let f be a classical holomorphic cusp form for SL2(Z) of weight k which is a normalized eigenfunction for the Hecke algebra, and let λ(n) be its eigenvalues. In this paper we study “shifted convolution sums” Σ n λ(n)λ(n + h) after averaging over many shifts h and obtain asymptotic estimates. The result is somewhat surprising: one encounters a transition region depending on the ratio of the square of the length of the average over h to the length of the shifted convolution sum. The phenomenon is similar to that encountered by Conrey, Farmer and Soundararajan in their 2000 paper [1] on transition mean values of the Jacobi symbol, and the connection of both results to Eisenstein series and multiple Dirichlet series is discussed.
| Type: | Article |
|---|---|
| Title: | Transition mean values of shifted convolution sums |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.jnt.2013.04.003 |
| Publisher version: | https://doi.org/10.1016/j.jnt.2013.04.003 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Shifted convolution sums, Modular forms, Eisenstein series, Multiple Dirichlet series |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10084831 |
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