Integral mean values of modular \(L\)-functions
type: Item
instance of: scholarly article
title: Integral mean values of modular \(L\)-functions
zbMATH Open document ID: 1091.11015
author: Qiao Zhang
published in: Journal of Number Theory
publication date: 2006-03-08T00:00:00Z
review text: The author considers the mean squares of \(L\)-functions associated to modular forms with respect to Hecke congruence subgroups, expressing the mean value as an inner product. This avoids the discussion of generalized additive divisor problems. More precisely, let \(f\in\mathcal S_k(N, \chi)\) be a cusp form for \(\Gamma_0(N)\) of weight \(k\) and character \(\chi\bmod N\). Let \(L(s, f)\) be associated \(L\)-function, normalized so that the central point is at \(s=1/2\). The author proves that \[ \int_0^T\left| L\left(\frac12+it, f\right)\right| ^2\,dt\sim\frac{2(4\pi)^k}{\Gamma(k)} \frac{\| f\| ^2}{\text{vol}(\Gamma_0(N)\backslash\mathfrak{H})}\,T\log T, \] where the norm on \(\mathcal S_k(N, \chi)\) is induced from the inner product \[ \langle f_1, f_2\rangle=\iint_{\Gamma_0(N)\backslash\mathfrak{H}} f_1(z)\overline{f_2(z)}y^k\,\frac{dx\,dy}{y^2}. \] An asymptotic formula for weighted mean square is also obtained. For the proof he considers the Dirichlet integral \[ Z_f(w)=\int_1^\infty\left| L\left(\tfrac12+it, f\right)\right| ^2t^{-w}\,dt \] and studies its analytic properties, especially its meromorphic continuation beyond \(\Re w=1\) and its polar behavior. The author points that this realizes and generalizes the ideas suggested by \textit{A. Good} [``The Selberg trace formula and related topics'', Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 53, 207--214 (1986; Zbl 0601.10020)].
reviewed by: Ramūnas Garunkštis
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 5009840
zbMATH Keywords: mean values
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: modular form
zbMATH Keywords: Hecke congruence subgroup
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.10.007
OpenAlex ID: W1984546215
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DOI: 10.1016/J.JNT.2004.10.007
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description: scientific article; zbMATH DE number 5009840