Benford’s Law for Coefficients of Newforms

Let f(z) = ∑∞ n=1 λf (n)e 2πinz ∈ S k (Γ0(N)) be a newform of even weight k ≥ 2 on Γ0(N) without complex multiplication. Let P denote the set of all primes. We prove that the sequence {λf (p)}p∈P does not satisfy Benford’s Law in any base b ≥ 2. However, given a base b ≥ 2 and a string of digits S in base b, the set Aλf (b, S) := {p prime : the first digits of λf (p) in base b are given by S} has logarithmic density equal to logb(1+S −1). Thus {λf (p)}p∈P follows Benford’s Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.