On the normal number of prime factors of sums of Fourier coefficients of eigenforms
type: Item
instance of: scholarly article
title: On the normal number of prime factors of sums of Fourier coefficients of eigenforms
zbMATH Open document ID: 1492.11089
author: M. Ram Murty
author: V. Kumar Murty
author: Sudhir Pujahari
published in: Journal of Number Theory
publication date: 2022-01-05T00:00:00Z
review text: In this work, the authors considered the problem of normal number of prime factors of sums of Fourier coefficients of eigenforms. They studied the normal number of prime factors of \(a_{f}(p)+a_{g}(p)\) with \(p\) prime, \(f\) and \(g\) distinct Hecke eigenforms of weight two. They proved that for any two independent normalized Hecke eigenforms \(f\) and \(g\) not of CM type with integer coefficients and of weight \(2\), assuming a quasi-generalized Riemann hypothesis \[ \sum_{p\leq x,a_{f}(p)+a_{g}(p)\neq 0}(\omega (a_{f}(p)+a_{g}(p))-\log \log x)^{2}=O(\pi (x)\log \log x), \] where the sum is over primes \(p\leq x\). Further, they proved that \[ \sum_{p\leq x}^{\prime } \omega (a_{p})=\pi (x)\log \log x+O(\pi (x)) \] and also \[ \sum_{p\leq x}^{\prime} \omega (a_{p})^{2}=\pi (x)(\log \log x)^{2}+O(\pi (x)\log \log x), \] where the dash on the sum indicates that sum is taken over those primes \(p\) for which \(a_p \neq 0\).
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 7452391
zbMATH Keywords: normal order
zbMATH Keywords: Chebotarev density theorem
zbMATH Keywords: Galois representations
Wikidata QID: Q114156929
reviewed by: Ahmet Tekcan
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.05.015
OpenAlex ID: W3186691387
cites work: Q5607047
cites work: Q4002278
cites work: Q5343492
cites work: Q4135671
cites work: IMAGES OF ADELIC GALOIS REPRESENTATIONS FOR MODULAR FORMS
cites work: Q3939877
cites work: Prime divisors of Fourier coefficients of modular forms
cites work: An analogue of the Erdős-Kac theorem for Fourier coefficients of modular forms
cites work: Distinguishing Hecke eigenforms
cites work: Modular Forms and the Chebotarev Density Theorem
cites work: Explicit formulae and the Lang-Trotter conjecture
cites work: Q4213367
cites work: On \(\ell\)-adic representations attached to modular forms
DOI: 10.1016/J.JNT.2021.05.015
description: scientific article; zbMATH DE number 7452391