On sums of Fourier coefficients of cusp forms
type: Item
instance of: scholarly article
title: On sums of Fourier coefficients of cusp forms
zbMATH Open document ID: 0696.10020
published in: L'Enseignement Mathématique. 2e Série
publication date: 1989-01-01T00:00:00Z
review text: Let \(F(z)=\sum a(n)e^{2\pi inz}\) be a cusp form of weight k for the full modular group, and set \(A(x)=\sum_{n\leq x}a(n)\). This paper is concerned with estimates for A(x) and for the error term B(x) in the asymptotic formula \[ \int^{x}_{0}A(t)^ 2 dt=Cx^{k+1/2}+B(x). \] It is shown that \(A(x)=O(x^{k/2-1/6})\) and \[ A(x)=\Omega_{\pm}(x^{k/2-1/4} \exp \{\frac{D(\log \log x)^{1/4}}{(\log \log \log x)^{3/4}}\}), \] for a suitable constant \(D>0\). Moreover B(x) satisfies the same \(\Omega\)-estimate except that \(``\Omega_{\pm}''\) must be replaced by ``\(\Omega\) ''.
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4139841
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: summatory function
zbMATH Keywords: cusp form
zbMATH Keywords: error term
zbMATH Keywords: asymptotic formula
zbMATH Keywords: \(\Omega\)-estimate
author: James Lee Hafner
author: Aleksandar Ivić
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4139841