Simultaneous non-vanishing of quadratic Dirichlet \(L\)-functions and twists of Hecke \(L\)-functions
type: Item
instance of: scholarly article
title: Simultaneous non-vanishing of quadratic Dirichlet \(L\)-functions and twists of Hecke \(L\)-functions
author: Gopal Maiti
published in: Mathematische Zeitschrift
publication date: 2022-12-08T00:00:00Z
review text: 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 \(q \to \infty\).
reviewed by: Qiao Zhang
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11M99
zbMATH DE Number: 7629768
zbMATH Keywords: simultaneous non-vanishing
zbMATH Keywords: \(L\)-functions
MaRDI profile type: Publication
full work available at URL: s00209-022-03166-6
OpenAlex ID: W4310676289
cites work: On moments of twisted L-functions
cites work: The second moment of twisted modular \(L\)-functions
cites work: The cubic moment of central values of automorphic \(L\)-functions
cites work: Simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass L-functions
cites work: A mean value estimate for real character sums
cites work: Q4830109
cites work: The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros
cites work: On the Mean Value of L(1/2, χ ) FW Real Characters
cites work: Simultaneous nonvanishing of 𝐺𝐿(2)×𝐺𝐿(2) and 𝐺𝐿(2) 𝐿-functions
cites work: Simultaneous nonvanishing of automorphic \(L\)-functions
cites work: Q4797985
cites work: A note on simultaneous nonvanishing twists
cites work: Average values of modular \(L\)-series via the relative trace formula
cites work: Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=\frac{1}{2}\)
cites work: A note on simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass $L$-functions
cites work: The second moment of quadratic twists of modular \(L\)-functions
cites work: Simultaneous nonvanishing of automorphic \(L\)-functions at the central point
cites work: Simultaneous non-vanishing for Dirichlet \(L\)-functions
DOI: 10.1007/S00209-022-03166-6
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description: scientific article; zbMATH DE number 7629768