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Finite Differences of the Logarithm of the Partition Function (1407.0177v1)

Published 1 Jul 2014 in math.NT and math.CO

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 $n$.

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