A note on the Fourier coefficients of Hecke-Maass forms
In the work the author studies the Hecke-Maass cusp forms over the full modular group \(\mathrm{SL}_{2}(\mathbb{Z})\). Letting \(f\) stand for a Maass form he uses the abbreviation \(F=\mathrm{sym}^{2} f\) and considers the symmetric square \(L\)-function attached to \(f\), \[ L(s,F)=\sum_{n=1}^{\infty}\frac{\lambda_{F}(n)}{n^{s}}. \] He then proves the following uniform upper bound for the sum over the coefficients \(\lambda_{F}(n)\): For any \(\varepsilon>0\) \[ \sum_{n\leq x}\lambda_{F}(n)\ll \nu^{3/4+\varepsilon}x^{1/2+\varepsilon}+x^{39/64+\varepsilon}, \tag{1} \] where the real parameter \(\nu\) arises from the non-euclidean Laplacian eigenvalue \(1/4+\nu^{2}\) of the corresponding Maass form. As a corollary he further attains the same upper bound for the following sum over the Hecke eigenvalues of a Maass form: For any \(\varepsilon>0\) \[ \sum_{n\leq x}\lambda_{f}(n^{2})\ll \nu^{3/4+\varepsilon}x^{1/2+\varepsilon}+x^{39/64+\varepsilon}. \tag{2} \] This type of sums have been investigated fairly much in the past, mainly in the case of holomorphic cusp forms. The author points out the recent results by \textit{G. Lü} [Acta Math. Hung. 124, No. 1--2, 83--97 (2009; Zbl 1200.11031)]. In the case of the sum (1) the author is able to improve the earlier results due to Lü when \(\nu\ll x^{43/216}\) or \(\nu\gg x^{2/3}\). In the case of (2) the author's bound is better everywhere. The proofs are based on careful argumentation using classical methods of analytic number theory, inspired by the work of \textit{Y. Ichihara} [Lith. Math. J. 48, No. 2, 188--202 (2008; Zbl 1143.11320)] in the case of holomorphic cusp forms.
scientific article; zbMATH DE number 6138261
scientific article