On Logarithmically Benford Sequences

Abstract

Let $\mathcal{I} \subset \mathbb{N}$ be an infinite subset, and let $\{a_i\}_{i \in \mathcal{I}}$ be a sequence of nonzero real numbers indexed by $\mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| \leq C_1 \cdot i^m$ for all $i \in \mathcal{I}$. Furthermore, let $c_i \in [-1,1]$ be defined by $c_i = \frac{a_i}{C_1 \cdot i^m}$ for each $i \in \mathcal{I}$, and suppose the $c_i$'s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $μ$. In this paper, we show that if $\mathcal{I} \subset \mathbb{N}$ is not too sparse, then the sequence $\{a_i\}_{i \in \mathcal{I}}$ fails to obey Benford's Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $μ([0,t])$ is a strictly convex function of $t \in (0,1)$. Nonetheless, we also provide conditions on the density of $\mathcal{I} \subset \mathbb{N}$ under which the sequence $\{a_i\}_{i \in \mathcal{I}}$ satisfies Benford's Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benford's Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.