Finite Differences of the Logarithm of the Partition Function
William Y.C. Chen, Larry X.W. Wang and Gary Y.B. Xie
Abstract: Let p(n) denote the partition function. DeSalvo and Pak proved that for n ≥ 2, as conjectured by Chen. Moreover, they conjectured that a sharper inequality holds for n ≥ 45. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for -Δ2log p(n - 1), where Δ is the difference operator with respect to n. We also show that for given r ≥ 1 and sufficiently large n, (-1)Δrlog 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 ≥ 1, Δrp(n) > 0 for sufficiently large n.
AMS Classification: 05A20, 11B68
Keywords: partition function, log-concavity, finite difference, the Lambert W function, the Hardy-Ramanujan-Rademacher formula