On fractional power moments of \(L\)-functions associated with certain cusp forms
Let \(f\) be a cusp form of weight \(\kappa\) for \(\text{SL}_2(\mathbb{Z})\), and let \(\phi(s, f)\) be the corresponding \(L\)-function (with \(\kappa/2\) being the centre of the critical strip). For a real number \(k \geq 0\) consider the moment \[ I_k(T, \sigma) = \int_0^T |\phi(\sigma + it, f)|^{2k}\,dt. \] The authors prove the following result: If \(k = p/q \leq 1/2\) is rational, then \(I_k(T, \kappa/2) \gg_k T(\log T)^{k^2}\). If in addition the Riemann hypothesis for \(\phi(s, f)\) is true, then also \(I_k(T, \kappa/2) \ll_k T(\log T)^{k^2}\). In the case \(p = 1\) this was proved by \textit{A. Laurinčikas} und \textit{J. Steuding} [Acta Appl. Math. 97, No. 1-3, 25--39 (2007; Zbl 1175.11049)] whose proof used ideas of \textit{D. R. Heath-Brown} [J. Lond. Math. Soc., II. Ser. 24, 65-78 (1981; Zbl 0431.10024)]. In the erratum Lemma 2.4 has been corrected.
scientific article; zbMATH DE number 5700608
scientific article