On Benford's Law for multiplicative functions

TL;DR

This paper establishes a criterion using Halász's Theorem to identify when real multiplicative functions follow Benford's Law strongly, revealing new classes of such functions and a group structure among non-Benford functions.

Contribution

It introduces a novel criterion based on Halász's Theorem for strong Benford behavior in multiplicative functions, expanding understanding of their distribution properties.

Findings

k-divisor functions with k ≠ 10^j are strong Benford

Hecke eigenvalues of newforms are strong Benford

Non-strong Benford multiplicative functions form a group

Abstract

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Hal\'{a}sz's Theorem.