The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II
type: Item
instance of: scholarly article
title: The additive divisor problem and its analogs for Fourier coefficients of cusp forms. II
zbMATH Open document ID: 0919.11059
DOI: 10.1007/PL00004323
author: Matti Jutila
published in: Mathematische Zeitschrift
publication date: 1999-04-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174930
review text: This is the continuation of the author's earlier work [Math. Z. 223, 435-461 (1996; Zbl 0865.11062)] on the analogue of the binary additive divisor problem for Fourier coefficients of cusp forms (holomorphic and non-holomorphic). The author gives now a detailed proof of the bound (stated already in Part I of the paper) \[ \sum_{\kappa_j\leq K}|c_j|^2\exp(\pi\kappa_j) + \int_{- K}^K|c(u)|^2\exp(\pi|u|) \text{ d}u \ll_{\varepsilon,\kappa} K^\varepsilon, \] which is important in treating the additive problem for Fourier coefficients of cusp forms. The above result is valid for \(K \geq 1\), and the notation is the same as in Part I: \[ c_j = (u_j(z),u^2(z)),\quad c(t) = \Bigl(E(z, {\textstyle{1\over 2}}+it),u^2(z)\Bigr), \] where \((f,g)\) is the Petersson inner product of \(f\) and \(g\), \(u(z)\) is a Maaas wave form of certain fixed index \(\nu\), \(\kappa = \kappa_{\nu}\), and \(E(z,s)\) is the non-holomorphic Eisenstein series. The intricate proof, based on three lemmas, involves techniques such as the use of Bessel functions, the hypergeometric function and the spectral large sieve inequality.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 1277931
zbMATH Keywords: Maass wave form
zbMATH Keywords: holomorphic cusp forms
zbMATH Keywords: Fourier series
zbMATH Keywords: Petersson inner product
zbMATH Keywords: spectral large sieve
zbMATH Keywords: non-holomorphic Eisenstein series
zbMATH Keywords: mean value estimate
zbMATH Keywords: additive divisor problem
zbMATH Keywords: Fourier coefficients of Maass forms
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1277931