The log-behavior of $$\root n \of {p(n)}$$ p ( n ) n and $$\root n \of {p(n)/n}$$ p ( n ) / n n

Research paper by William Y. C. Chen, Ken Y. Zheng

Indexed on: 13 Nov '16Published on: 01 Nov '16Published in: The Ramanujan Journal

Abstract

Abstract Let p(n) denote the partition function and let $$\Delta$$ be the difference operator with respect to n. In this paper, we obtain a lower bound for $$\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}$$ , leading to a proof of a conjecture of Sun on the log-convexity of $$\{\root n \of {p(n)/n}\}_{n\ge 60}$$ . Using the same argument, it can be shown that for any real number $$\alpha$$ , there exists an integer $$n(\alpha )$$ such that the sequence $$\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}$$ is log-convex. Moreover, we show that $$\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}$$ . Finally, by finding an upper bound for $$\Delta ^2 \log \root n-1 \of {p(n-1)}$$ , we establish an inequality on the ratio $$\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}$$ .AbstractLet p(n) denote the partition function and let $$\Delta$$ be the difference operator with respect to n. In this paper, we obtain a lower bound for $$\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}$$ , leading to a proof of a conjecture of Sun on the log-convexity of $$\{\root n \of {p(n)/n}\}_{n\ge 60}$$ . Using the same argument, it can be shown that for any real number $$\alpha$$ , there exists an integer $$n(\alpha )$$ such that the sequence $$\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}$$ is log-convex. Moreover, we show that $$\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}$$ . Finally, by finding an upper bound for $$\Delta ^2 \log \root n-1 \of {p(n-1)}$$ , we establish an inequality on the ratio $$\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}$$ .pn $$\Delta$$ $$\Delta$$n $$\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}$$ $$\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}$$ $$\{\root n \of {p(n)/n}\}_{n\ge 60}$$ $$\{\root n \of {p(n)/n}\}_{n\ge 60}$$ $$\alpha$$ $$\alpha$$ $$n(\alpha )$$ $$n(\alpha )$$ $$\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}$$ $$\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}$$ $$\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}$$ $$\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}$$ $$\Delta ^2 \log \root n-1 \of {p(n-1)}$$ $$\Delta ^2 \log \root n-1 \of {p(n-1)}$$ $$\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}$$ $$\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}$$