Power moments of automorphic \(L\)-function attached to Maass forms (Q2789373)

The paper under review considers the power moments of the automorphic \(L\)-function \(L(s,f)\) of a normalized Maass cusp form \(f\) for the modular group \(\mathrm{SL}_2(\mathbb{Z})\). Let \(\sigma\) be a fixed real number such that \(1/2<\sigma <1\). Let \(m(\sigma)\) be the supremum of all numbers \(m\geq 2\) such that NEWLINE\[NEWLINE \int_1^T |L(\sigma +it)|^m dt \ll_{f,\varepsilon} T^{1+\varepsilon}, NEWLINE\]NEWLINE where the implied constant depends on \(f\) and \(\varepsilon >0\). The main result are the lower bounds for \(m(\sigma)\). More precisely, it is proved that NEWLINE\[NEWLINE m(\sigma)\geq \frac{2}{3-4\sigma}, \quad \text{for} \quad \frac{1}{2}<\sigma\leq\frac{5}{8} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE m(\sigma)\geq \frac{5-2\sigma}{2(1-\sigma)(3-2\sigma)}, \quad \text{for} \quad\frac{5}{8}<\sigma \leq 1-\varepsilon. NEWLINE\]NEWLINE For \(\frac{1}{2}<\sigma\leq\frac{5}{8}\), the same bound in the case of holomorphic cusp forms is obtained by \textit{A. Ivić} [in: Proceedings of the Amalfi conference on analytic number theory, held at Maiori, Italy, 1989. Salerno: Universitá di Salerno. 231--246 (1992; Zbl 0787.11035)]. The main difficulty in the case of Maass cusp forms is that the Ramanujan conjecture is not known, and the bounds of \textit{H. H. Kim} and \textit{P. Sarnak} from Appendix 2 to [\textit{H. H. Kim}, J. Am. Math. Soc. 16, No. 1, 139--183 (2003; Zbl 1018.11024)] are used instead. As a consequence of the lower bounds, asymptotic formulas for the second, fourth and sixth power moments of \(L(s,f)\) are obtained.