Benford’s Law for coefficients of newforms

Abstract

Let [Formula: see text] be a newform of even weight [Formula: see text] on [Formula: see text] without complex multiplication. Let [Formula: see text] denote the set of all primes. We prove that the sequence [Formula: see text] does not satisfy Benford’s Law in any integer base [Formula: see text]. However, given a base [Formula: see text] and a string of digits [Formula: see text] in base [Formula: see text], the set [Formula: see text] has logarithmic density equal to [Formula: see text]. Thus, [Formula: see text] follows Benford’s Law with respect to logarithmic density. Both results rely on the now-proven Sato–Tate Conjecture.