On the Digits of Partition Functions

TL;DR

This paper investigates the initial digits of the partition function p(n) in a given base, providing improved bounds on the smallest n where p(n) starts with a specific digit string.

Contribution

It introduces an elementary discrepancy approach to derive sharper upper bounds on the initial digits of p(n), advancing prior work by Luca.

Findings

Established new upper bounds for the smallest n with specified starting digits in p(n).

Improved upon previous bounds by Luca using discrepancy methods.

Enhanced understanding of the digit distribution in partition functions.

Abstract

We study a problem of Douglass and Ono concerning the smallest integer $n$ such that the partition function $p(n)$ begins with a specified string of digits $f$ in base $b$. By employing an elementary discrepancy framework, we establish new upper bounds that significantly improve upon previous results of Luca.