The mean square of Dirichlet series associated with automorphic forms
type: Item
instance of: scholarly article
title: The mean square of Dirichlet series associated with automorphic forms
zbMATH Open document ID: 0773.11033
DOI: 10.1007/BF01303063
author: S. Singh
published in: Monatshefte für Mathematik
publication date: 1993-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/178557
review text: The author considers real analytic automorphic forms of real weight \(k\), multiplier system \(\chi\) and eigenvalue \(\lambda={1\over 4}-\rho^ 2\) \((\text{Re} \rho\geq 0)\). Theorem 4.1 is an extension of the well-known Hecke-Maaß correspondence between automorphic forms and pairs of Dirichlet series \(L^ \pm(f,s)\) with functional equation. The main results of the work under review are mean square estimates for the Dirichlet series \(L^ \pm(f,s)\). If \(f\) is not a cusp form, the author proves that \[ \int^ T_ 0| L^ \pm(f,\text{Re} \rho+it)|^ 2\;dt=T(\log T)^ \alpha(B^ \pm+O((\log T)^{-1})) \] where \(B^ \pm\) are constants and \(\alpha\) is either 1, 2 or 4, provided that the underlying group \(\Gamma\) is a congruence subgroup of \(SL_ 2(\mathbb{Z})\). If \(\Gamma\) is a finitely generated Fuchsian group of the first kind with cusps \(\infty,0\), the same result holds with \(o(1)\) instead of \(O((\log T)^{-1})\), and there are similar results for cusps forms. These results are applied to the Dirichlet series associated with theta functions corresponding to positive definite or indefinite quadratic forms, that is, to Epstein zeta-functions and Siegel zeta-functions. In the case of the Riemann zeta-function, the author deduces Ingham's fourth power moment estimate from his result. (The reviewer is angry at the fact that in this work on real analytic automorphic forms the names of H. Maaß and W. Roelcke appear only in wrong orthography -- and even A. Selberg lost his ``g'' at least once).
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 164940
zbMATH Keywords: extension of Hecke-Maaß correspondence
zbMATH Keywords: real analytic automorphic forms
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Dirichlet series
zbMATH Keywords: functional equation
zbMATH Keywords: mean square estimates
zbMATH Keywords: Epstein zeta-functions
zbMATH Keywords: Siegel zeta-functions
zbMATH Keywords: Ingham's fourth power moment estimate
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
cites work: Q5558293
cites work: Kloosterman sums and Fourier coefficients of cusp forms
cites work: Q3220660
cites work: Q3213184
cites work: Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen. (The differential equations in the theory of elliptic modular functions)
cites work: The mean square of the Dedekind zeta function in quadratic number fields
cites work: Q4154638
cites work: Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II. (The eigenvalue problem of automorphic forms in the hyperbolic plane. I. II)
cites work: Über die Zetafunktionen indefiniter quadratischer Formen. II
cites work: Indefinite quadratische Formen und Funktionentheorie. I
cites work: Indefinite quadratische Formen und Funktionentheorie. II
cites work: Q5790818
OpenAlex ID: W2024358196
description: scientific article; zbMATH DE number 164940