Finite Differences of the Logarithm of the Partition Function

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 $n\ge 2$, 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 $n\ge 45$. 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 $r\ge 1$ 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 $r\ge 1$, ${\Delta }^{r}p(n)>0$ for sufficiently large $n$.

28 pages