Distribution of Coefficients of Modular Forms and the Partition Function

TL;DR

This paper investigates the distribution of modular form coefficients modulo odd integers and applies results to the partition function, demonstrating a lower bound on the frequency of partition numbers in residue classes.

Contribution

It establishes new distribution results for modular form coefficients modulo odd integers and derives implications for the partition function's residue class distribution.

Findings

Distribution of modular form coefficients modulo odd integers is well-understood.

Partition function values are evenly distributed among residue classes with a specific lower bound.

Provides quantitative bounds on the frequency of partition numbers in residue classes.

Abstract

Let $\ell\ge5$ be an odd prime and $j, s$ be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo odd positive integer $M$. As a consequence, we prove that for each integer $1\le r\le\ell^j$, $$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j}\}\gg_{s,r,\ell^j}\frac{\sqrt X}{\log X}(\log\log X)^s.$$