Lower bound for higher moments of the mixed product of twisted \(L\)-functions
Let $\mathcal F(\kappa)$ stand for the set of the holomorphic cuspidal Hecke eigenforms \[f(z):=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{(\kappa-1)/2}e(nz),\; z\in\mathbb{C},\;\text{Im}\;(z) >0,\] of level $1$ and weight $\kappa$ and let, for $f\in\mathcal F(\kappa)$ and a Dirichlet character $\chi$, \[L(f, \chi; s):=\sum_{n=1}^{\infty}\lambda_{f}(n)\chi(n)n^{-s},\;\text{Re}\;(s) >1.\] Suppose that $f_{1}\in\mathcal F(\kappa_{1}), f_{2}\in\mathcal F(\kappa_{2}),$ and $f_{1}\neq f_{2}$; let $ q\in\mathbb{N},\; q\not\equiv 2\;\text{mod}\;4$, $q=q_{0}q_{1},\;\{q_{0}, q_{1}\}\subset\mathbb{N},\;q_{1}$ be odd, and $q^{\eta}<<q_{0}<<q^{1/2 - \eta}$ for a real number $\eta$ in the interval $0<\eta <1/2$. The authors prove then that \[\sum_{\chi\;\text{mod}\;q}( L(f_{1}, \chi; 1/2)\overline{ L(f_{2}, \chi; 1/2)})^{2k}>>_{k} \psi (q) (\log q)^{2k^{2}} (\log\log q)^{-4k^{2}},\;k\in\mathbb{N},\] where summation is restricted to the primitive characters $\chi$ modulo $q$. According to the authors, to obtain their lower bound they use the methods of the works of \textit{Z. Rudnick} and \textit{K. Soundararajan} [Proc. Natl. Acad. Sci. USA 102, No. 19, 6837--6838 (2005; Zbl 1159.11317); Proc. Symp. Pure Math. 75, 293--303 (2006; Zbl 1120.11039)] and an idea of \textit{V. Blomer} and \textit{D. Milićević} [Geom. Funct. Anal. 25, No. 2, 453--516 (2015; Zbl 1400.11097)].
scientific article; zbMATH DE number 7318736
scientific article