Benford's Law for Coefficients of Newforms

Abstract

Let $f(z)={\sum }_{n=1}^{\infty }{\lambda }_f(n)e^{2\pi inz}\in S_k^{new}({\Gamma }_0(N))$ be a normalized Hecke eigenform of even weight $k\ge 2$ on ${\Gamma }_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence $\{{\lambda }_f(p){\}}_{p\in \mathbb{P}}$ does not satisfy Benford's Law in any base $b\ge 2$. However, given a base $b\ge 2$ and a string of digits $S$ in base $b$, the set \[ A_{λ_f}(b,S):=\{\text{$p$ prime : the first digits of ${\lambda }_f(p)$ in base $b$ are given by $S$}\} \] has logarithmic density equal to ${\log }_b(1+S^{-1})$. Thus $\{{\lambda }_f(p){\}}_{p\in \mathbb{P}}$ follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.

10 pages. Referee comments implemented. To appear in International Journal of Number Theory

Cited by in corpus (1)

• On Logarithmically Benford Sequences