Log-Concavity of the Partition Function

TL;DR

This paper proves the log-concavity of the partition function p(n) for all n > 25 and extends these results to address related conjectures, using advanced estimates of series remainders.

Contribution

It establishes the log-concavity of p(n) beyond n=25 and resolves two conjectures by Chen through novel analytical techniques.

Findings

p(n) is log-concave for all n > 25

Extended results to two related conjectures by Chen

Used Lehmer's estimates on Hardy-Ramanujan and Rademacher series

Abstract

We prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the Rademacher series for $p(n)$.