Finite Differences of the Logarithm of the Partition Function

2014/07/01 by Chen, William Y. C., Wang, Larry X. W., Xie, Gary Y. B. · 1 citation
#05A20 #11B68 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1407.0177

Abstract

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, as conjectured by Chen. Moreover, they conjectured that a sharper inequality (p(n-1))/(p(n))( 1+\fracπ√(24)n^3/2) > (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 -Δ² log 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)^r-1Δ^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, Δ^r p(n)>0 for sufficiently large n.

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