Benford's Law for Coefficients of Newforms

TL;DR

This paper investigates the digit distribution of Fourier coefficients of newforms, showing they follow Benford's Law with respect to logarithmic density, relying on the Sato-Tate Conjecture.

Contribution

It proves the digit distribution of newform coefficients follows Benford's Law in terms of logarithmic density, a novel connection between number theory and digit analysis.

Findings

Coefficients do not satisfy Benford's Law in the usual sense.

Distribution of leading digits follows Benford's Law with respect to logarithmic density.

Results depend on the proven Sato-Tate Conjecture.

Abstract

Let $f(z)=\sum_{n=1}^\infty \lambda_f(n)e^{2\pi i n z}\in S_{k}^{new}(\Gamma_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ 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 $b\geq2$. However, given a base $b\geq2$ and a string of digits $S$ in base $b$, the set \[ A_{\lambda_f}(b,S):=\{\text{$p$ prime : the first digits of $\lambda_f(p)$ in base $b$ are given by $S$}\} \] 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.