Coprimality of Fourier coefficients of eigenforms

Abstract

1.1. Motivation and the first result. Given two integer-valued sequences a1(n) and a2(n), an interesting question is, how the sequence of the gcd’s (a1(n), a2(n)) behaves; and in particular, how often a1(n) and a2(n) are coprime. For example, if a1(n) = n and a2(n) = φ(n), the Euler’s φ-function, then the density of such integers is zero. This follows from the beautiful result of Erdős [Erd49] given below: