论文标题
不平等的不平等差异差异的差异和王Xie-zhang的问题
Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang-Xie-Zhang
论文作者
论文摘要
令$ \ overline {p}（n）$表示分支机构功能。在本文中，我们的主要目标是研究过度函数对数有限差异的渐近行为，即$（-1）^{r-1}Δ^r \ log \ log \ log \ p（n）$，通过研究以下形式的不平等\ bigl（1+ \ dfrac {c（r）} {n^{n^{r-1/2}} - \ dfrac {c_1（r）} {n^{r}}} \ bigr）<（-1）<（-1） \Bigl(1+\dfrac{C(r)}{n^{r-1/2}}\Bigr)\ \text{for}\ n \geq N(r),$$ where $C(r), C_1(r), \text{and}\ N(r)$ are computable constants depending on the positive integer $r$, determined explicitly.这解决了Wang，Xie和Zhang在搜索$（-1）^{r-1}Δ^r \ log \ p（n）$的更好的下限的上下文中提出的问题。通过解决问题，我们可以证明\ begin {equation*} \ lim_ {n \ rightarrow \ infty}（ - 1）^{r-1}δ^r \ log \ log \ log \ p（n） = \dfracπ{2} \ bigl（\ dfrac {1} {2} {2} \ bigr）_ {r-1} n^{\ frac {1} {2} {2} -r}。 \ end {equation*}
Let $\overline{p}(n)$ denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., $(-1)^{r-1}Δ^r \log \p(n)$, by studying the inequality of the following form $$\log \Bigl(1+\dfrac{C(r)}{n^{r-1/2}}-\dfrac{C_1(r)}{n^{r}}\Bigr)<(-1)^{r-1}Δ^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 $(-1)^{r-1}Δ^r \log \p(n)$ than $0$. By settling the problem, we are able to show that \begin{equation*} \lim_{n\rightarrow \infty}(-1)^{r-1}Δ^r \log \p(n) =\dfracπ{2}\Bigl(\dfrac{1}{2}\Bigr)_{r-1}n^{\frac{1}{2}-r}. \end{equation*}
