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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 \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\) \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\) \(\{\root n \of {p(n)/n}\}_{n\ge 60}\) \(\{\root n \of {p(n)/n}\}_{n\ge 60}\) \(\alpha \) \(\alpha \) \(n(\alpha )\) \(n(\alpha )\) \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\) \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\) \(\Delta ^2 \log \root n-1 \of {p(n-1)}\) \(\Delta ^2 \log \root n-1 \of {p(n-1)}\) \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\) \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\)