Title: On the Stieltjes Approximation Error to Logarithmic Integral Show Chinese title

Title: 关于对数积分的Stieltjes逼近误差

Abstract: This paper studies the error introduced ($\displaystyle \varepsilon(x)$) by the Stieltjes asymptotic approximation series $\displaystyle li_{*}(x) = \frac{x}{\log(x)}\sum_{k=0}^{n-1}\frac{k!}{\log^{k}(x)} + (\log(x)-n)\frac{xn!}{\log^{n+1}(x)}$ to the logarithmic integral function $li(x)$ with $\displaystyle n = \lfloor \log(x) \rfloor$ for all $x\ge e$. For this purpose, this paper uses some relations between term $\displaystyle \frac{n!}{\log^{n}(x)}$ and Stirling's approximation formula. In particular, this paper establishes two non-asymptotic lower and upper bounds for $\varepsilon(e^n)$ with $n \ge 1$ and $1 \le m \le n$: (i) $\displaystyle \varepsilon(e^m) + \frac{\sqrt{2\pi}}{8}\sum_{k=m+1}^{n} \frac{1}{k^{\frac{3}{2}}} \le \varepsilon(e^n) \le \varepsilon(e^{m}) + \frac{\sqrt{2\pi}}{4}\sum_{k=m+1}^{n} \frac{k+2}{k^{\frac{5}{2}}}$ (ii) $\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} \le \varepsilon_{n} \le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$. Here, $\displaystyle L_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{4\sqrt{m+1}}$ and $\displaystyle R_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{2\sqrt{m}} + \frac{\sqrt{2\pi}}{3\sqrt{m^3}}$ Moreover, this paper shows that if $\displaystyle |\varepsilon(x)| \le \frac{\sqrt{2\pi}}{\sqrt{\log(x)}}$ then $li(e^n) \le li_{*}(e^n)$ for all $n \ge 1$. Finally, this paper establishes non-asymptotic lower and upper bounds for $\varepsilon(x)$ with $x \ge e$: $\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} -\frac{\sqrt{2\pi}}{36\sqrt{n^3}} < \varepsilon(x) \le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$ Show Chinese abstract

Abstract: 本文研究了斯特尔吉斯渐近展开式$\displaystyle li_{*}(x) = \frac{x}{\log(x)}\sum_{k=0}^{n-1}\frac{k!}{\log^{k}(x)} + (\log(x)-n)\frac{xn!}{\log^{n+1}(x)}$对对数积分函数$li(x)$的误差（$\displaystyle \varepsilon(x)$），其中$\displaystyle n = \lfloor \log(x) \rfloor$对所有$x\ge e$都成立。 为此，本文使用了项$\displaystyle \frac{n!}{\log^{n}(x)}$与斯特林近似公式之间的某些关系。 特别是，本文建立了针对$\varepsilon(e^n)$的两个非渐近下界和上界，分别为$n \ge 1$和$1 \le m \le n$: (i)$\displaystyle \varepsilon(e^m) + \frac{\sqrt{2\pi}}{8}\sum_{k=m+1}^{n} \frac{1}{k^{\frac{3}{2}}} \le \varepsilon(e^n) \le \varepsilon(e^{m}) + \frac{\sqrt{2\pi}}{4}\sum_{k=m+1}^{n} \frac{k+2}{k^{\frac{5}{2}}}$ (ii)$\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} \le \varepsilon_{n} \le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$ 这里，$\displaystyle L_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{4\sqrt{m+1}}$ 和 $\displaystyle R_m = \varepsilon_{m} + \frac{\sqrt{2\pi}}{2\sqrt{m}} + \frac{\sqrt{2\pi}}{3\sqrt{m^3}}$ 此外，本文表明如果 $\displaystyle |\varepsilon(x)| \le \frac{\sqrt{2\pi}}{\sqrt{\log(x)}}$ 则 $li(e^n) \le li_{*}(e^n)$ 对所有 $n \ge 1$ 成立。 最后，本文为$\varepsilon(x)$建立了非渐近下界和上界，其中$x \ge e$：$\displaystyle L_m - \frac{\sqrt{2\pi}}{4\sqrt{n+1}} -\frac{\sqrt{2\pi}}{36\sqrt{n^3}} < \varepsilon(x) \le R_m - \frac{\sqrt{2\pi}}{2\sqrt{n}} - \frac{\sqrt{2\pi}}{3\sqrt{n^3}}$