Benford's Law for Coefficients of Newforms

Abstract

Let $f(z)=\sum_{n=1}^\infty λ_f(n)e^{2πi n z}\in S_{k}^{new}(Γ_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ on $Γ_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence $\{λ_f(p)\}_{p\in\mathbb{P}}$ does not satisfy Benford's Law in any base $b\geq2$. However, given a base $b\geq2$ and a string of digits $S$ in base $b$, the set \[ A_{λ_f}(b,S):=\{\text{$p$ prime : the first digits of $λ_f(p)$ in base $b$ are given by $S$}\} \] has logarithmic density equal to $\log_b(1+S^{-1})$. Thus $\{λ_f(p)\}_{p\in\mathbb{P}}$ follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.