On sums of Hecke series in short intervals. (English) Zbl 0994.11020

Ivić, Aleksandar

J. Théor. Nombres Bordx. 13, No. 2, 453-468 (2001).

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 \(H_j(1/2)\ll \kappa_j^{1/3+\varepsilon}\).

Reviewer: Roger Heath-Brown (Oxford)

MSC:

11F66	Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F37	Forms of half-integer weight; nonholomorphic modular forms

Keywords:

cubic moment; short interval; Hecke series; Maass wave forms; upper bound

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