Benford's Law For Coefficients of Modular Forms and Partition Functions

TL;DR

This paper proves that the coefficients of certain modular forms and partition functions follow Benford's law, using asymptotic analysis based on harmonic Maass forms.

Contribution

It extends Benford's law to a broad class of modular form coefficients and partition functions through new asymptotic formulas.

Findings

Partition functions like p(n) follow Benford's law.

Coefficients of modular forms exhibit Benford's distribution.

Asymptotic formulas underpin the proof.

Abstract

Here we prove that Benford's law holds for coefficients of an infinite class of modular forms. Expanding the work of Bringmann and Ono on exact formulas for harmonic Maass forms, we derive the necessary asymptotics. This implies that the unrestricted partition function $p(n)$, as well as other natural partition functions, satisfy Benford's law.