Finite Differences of the Logarithm of the Partition Function

TL;DR

This paper proves a conjectured inequality involving the partition function by analyzing finite differences of its logarithm, establishing new bounds and positivity properties that deepen understanding of partition function behavior.

Contribution

The paper confirms a conjecture by Desalvo and Pak, providing bounds on second differences of the log of the partition function and demonstrating positivity of higher order differences for large n.

Findings

Proved the conjecture of Desalvo and Pak regarding inequalities of p(n) ratios.

Established upper bounds for second differences of log p(n).

Showed higher order differences of log p(n) are positive for large n.

Abstract

Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$ for $n\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality $\frac{p(n-1)}{p(n)}\left( 1+\frac{\pi}{\sqrt{24}n^{3/2}}\right) > \frac{p(n)}{p(n+1)}$ holds for $n\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-\Delta^{2} \log p(n-1)$, where $\Delta$ is the difference operator with respect to $n$. We also show that for given $r\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\Delta^{r} \log p(n)>0$. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given $r\geq 1$, $\Delta^{r} p(n)>0$ for sufficiently large $n$.