On the normal number of prime factors of sums of Fourier coefficients of eigenforms
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 \(a_p \neq 0\).
scientific article; zbMATH DE number 7452391
scientific article