an:01770781
Zbl 0994.11020
Ivi??, Aleksandar
On sums of Hecke series in short intervals
EN
J. Th??or. Nombres Bordx. 13, No. 2, 453-468 (2001).
00077685
2001
j
11F66 11F37
cubic moment; short interval; Hecke series; Maass wave forms; upper bound
Let \(H_j(s)\) be the standard set of Hecke series associated with Maass wave forms of the full modular group with associated eigenvalues \(\lambda_j= \kappa_j^2+ 1/4\). Then, with the usual notation, it is shown that
\[
\sum_{K-G< \kappa_j\leq K+G} \alpha_j H_j(1/2)^3\ll GK^{1+\varepsilon},
\]
for \(1\leq G\leq K\) and any fixed \(\varepsilon> 0\). As a corollary one has the new upper bound \(H_j(1/2)\ll \kappa_j^{1/3+\varepsilon}\).
Roger Heath-Brown (Oxford)