On the coefficients of cusp forms
type: Item
instance of: scholarly article
title: On the coefficients of cusp forms
zbMATH Open document ID: 0883.11020
author: Dinakar Ramakrishnan
published in: Mathematical Research Letters
publication date: 1998-03-05T00:00:00Z
review text: Fix a cuspidal representation \(\pi\) of \(GL(2)\) over (the adele ring of) a number field \(F\), and write \(a_v\) for the sum of the Hecke eigenvalues of \(\pi\) at a place \(v\) of \(F\) where \(\pi\) is unramified. Ramanujan's conjecture for \(\pi\) is equivalent to \(|a_v|\leq 2\) for all \(v\). It is shown that the lower Dirichlet density of the set of \(v\) with \(|a_v|\leq 2\) is at least \(0.9\) (one needs \(1\) for the conjecture), and that for \(\pi\) on \(GL(n)\) the set of \(v\) with \(|a_v|\leq n\) has such density \(\geq 1-1/n^2\) (but this is not directly related to Ramanujan's). Also, for \(n=2\), given \(\pi\), \(\epsilon>0\), \(m>1\), there exists a set \(S\) of primes of density \(1\) such that the partial \(L\)-function \(L_S(s,\pi,\text{sym}^m)\) converges on \(\text{Re} (s)> 1+\epsilon\).
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 1053510
zbMATH Keywords: Dirichlet density
zbMATH Keywords: Hecke eigenvalues
zbMATH Keywords: cuspidal representation
zbMATH Keywords: Ramanujan conjecture
zbMATH Keywords: partial \(L\)-function
reviewed by: Yuval Z. Flicker
MaRDI profile type: Publication
full work available at URL: mrl.1997.v4.n2.a10
OpenAlex ID: W2091350265
DOI: 10.4310/MRL.1997.V4.N2.A10
description: scientific article; zbMATH DE number 1053510