Simultaneous non-vanishing of quadratic Dirichlet \(L\)-functions and twists of Hecke \(L\)-functions
Let \(f \in S_k(\operatorname{SL}_2(\mathbb{Z}))\) be a Hecke eigenform. This paper considers the problem of simultaneous non-vanishing of the central \(L\)-values \(L(1 / 2, \chi_D)\) and \(L(1 / 2, f \otimes \chi_D)\) as \(D\) runs through fundamental discriminants, where \(\chi_D\) is the corresponding primitive quadratic Dirichlet character. For technical simplicity, the author confines his discussions to the case that the weight \(k\) is a multiple of 4, and that the fundamental discriminant is of form \(D = 8 d\) for some odd square-free positive integer \(d\). In this case, the author shows that \[ \sum_{\substack{0 < 8 d \leq X \\ \text{\(d\) odd square-free} \\ L(1/2, \chi_{8d}) L(1/2, f \otimes \chi_{8d}) \neq 0}} 1 \ \ \gg X^{2/3-\varepsilon} \] More precisely, the author derives the asymptotic formula for the first smoothed moment of \(L(1/2, \chi_{8d}) L(1/2, f \otimes \chi_{8d})\), as well as an upper bound for its second moment, and the above estimate follows by applying the Cauchy-Schwarz inequality. To establish his results, the author starts with the approximate functional equations of the two \(L\)-functions, and applies the Poisson summation formula and the large sieve inequality of \textit{D. R. Heath-Brown} [Acta Arith. 72, No. 3, 235--275 (1995; Zbl 0828.11040)]. The results in this paper may be compared with the work of \textit{Q. Sun} [J. Ramanujan Math. Soc. 37, No. 1, 31--47 (2022; Zbl 1494.11049)] which studies the simultaneous non-vanishing of the \(L\)-values \(L(1/2, \chi)\) and \(L(1/2, f \otimes \chi)\) as \(\chi\) runs through Dirichlet characters of a fixed conductor \(q\) with \(q \to \infty\).
scientific article; zbMATH DE number 7629768
scientific article