The Log-Behavior of √[n]p(n) and √[n]p(n)/n

2015/11/09 by Chen, William Y. C., Zheng, Ken Y.
#05A20 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1511.02558

Abstract

Let p(n) denote the partition function. Desalvo and Pak proved the log-concavity of p(n) for n>25 and the inequality (p(n-1))/(p(n))(1+(1)/(n))>(p(n))/(p(n+1)) for n>1. Let r(n)=√[n]p(n)/n and Δ be the difference operator respect to n. Desalvo and Pak pointed out that their approach to proving the log-concavity of p(n) may be employed to prove a conjecture of Sun on the log-convexity of \r(n)\_{n≥ 61}, as long as one finds an appropriate estimate of Δ² log r(n-1). In this paper, we obtain a lower bound for Δ²log r(n-1), leading to a proof of this conjecture. From the log-convexity of \r(n)\_n≥61 and \√[n]n\_n≥4, we are led to a proof of another conjecture of Sun on the log-convexity of \√[n]p(n)\_n≥27. Furthermore, we show that lim_{n → +∞}n^(5)/(2)Δ²log√[n]p(n)=3π/√(24). Finally, by finding an upper bound of Δ² log√[n-1]p(n-1), we prove an inequality on the ratio \frac√[n-1]p(n-1)√[n]p(n) analogous to the above inequality on the ratio (p(n-1))/(p(n)).

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