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 $n\geq 2$. 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 $n$.