Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang

TL;DR

This paper investigates the asymptotic behavior of higher order differences of the logarithm of the overpartition function, providing explicit inequalities and solving a problem related to lower bounds posed by Wang, Xie, and Zhang.

Contribution

It establishes explicit inequalities for the finite differences of the overpartition function's logarithm and solves an open problem about lower bounds, with precise asymptotic limits.

Findings

Derived explicit bounds for the differences of log overpartition function

Solved the problem of finding a better lower bound for these differences

Established the asymptotic limit of the differences as n approaches infinity

Abstract

Let $\overline{p}(n)$ denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., $(-1)^{r-1}\Delta^r \log \p(n)$, by studying the inequality of the following form $$\log \Bigl(1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{C_1(r)}{n^{r}}\Bigr)<(-1)^{r-1}\Delta^r \log \p(n) <\log \Bigl(1+\dfrac{C(r)}{n^{r-1/2}}\Bigr)\ \text{for}\ n \geq N(r),$$ where $C(r), C_1(r), \text{and}\ N(r)$ are computable constants depending on the positive integer $r$, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of $(-1)^{r-1}\Delta^r \log \p(n)$ than $0$. By settling the problem, we are able to show that \begin{equation*} \lim_{n\rightarrow \infty}(-1)^{r-1}\Delta^r \log \p(n) =\dfrac{\pi}{2}\Bigl(\dfrac{1}{2}\Bigr)_{r-1}n^{\frac{1}{2}-r}. \end{equation*}