FINITE DIFFERENCES OF THE LOGARITHM OF THE PARTITION FUNCTION

Let p(n) denote the partition function. DeSalvo and Pak proved that p(n-1)/p(n) (1 + 1/n) > p(n)/p(n+1) for n >= 2. Moreover, they conjectured that a sharper inequality p(n-1)/p(n) (1 + pi/root 24n3/2) > p(n)/p(n+1) holds for n >= 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 >= 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 >= 1, Delta(r)p(n) > 0 for sufficiently large n.