An estimate for symmetric square \(L\)-functions on weight aspect
Let \(B_k\) be the set of all normalized Hecke eigencuspforms for the full modular group. For \(f\in B_k\) let \(L(\text{sym}^2 f, s)\) be the symmetric square \(L\)-function of \(f\). Under the assumption that the central value \(L(\text{sym}^2 f, 1/2)\geq0\) for all \(f\in B_k\) and all \(k\), \textit{W. Kohnen} and \textit{J. Sengupta} [Proc. Am. Math. Soc. 128, 1641--1646 (2000; Zbl 1029.11015)] showed that for any \(\varepsilon>0\) and for all sufficiently large \(k\), \[ \sum_{f\in B_k}L(\text{sym}^2 f, 1/2)\ll_\varepsilon k^{1+\varepsilon}.\tag{\(*\)} \] Here the author proves that for any \(\varepsilon>0\), any real \(t\), and for all sufficiently large \(K\), \[ \sum_{\substack{ K\leq k\leq 2K\\ k\equiv 0\,(2)}} \sum_{f\in B_k}| L(\text{sym}^2 f, 1/2+it)| ^2 \ll_\varepsilon K^{2+\varepsilon}(| t| +1)^{3+\epsilon}. \] This yields that the bound (\(*\)) is unconditionally true for infinitely many even \(k\).
scientific article; zbMATH DE number 2051589