Defining Upper and Lower Bounding Functions of $li(x)$ with $\displaystyle O\left(\sqrt{\frac{x}{\log(x)}}\right)$ Error Using Truncated Asymptotic Series

Abstract

We introduce approximation functions of $li(x)$ for all $x\ge e$: (1) $\displaystyle li_{\underlineω,α}(x) = \frac{x}{\log(x)}\left( α\frac{\underline{m}!}{\log^{\underline{m}}(x)} + \sum_{k=0}^{\underline{m}-1}\frac{k!}{\log^{k}(x)} \right)$, and (2) $\displaystyle li_{\overlineω,β}=\frac{x}{\log(x)}\left( β\frac{\overline{m}!}{\log^{\overline{m}}(x)} + \sum_{k=0}^{\overline{m}-1}\frac{k!}{\log^{k}(x)} \right)$ with $0 < ω< 1$ a real number, $α\in \{ 0, \underlineκ\log(x) \}$, $\underline{m} = \lfloor \underlineκ\log(x) \rfloor$, $β\in \{ \overlineκ\log(x), 1 \}$, $\overline{m} = \lfloor \overlineκ\log(x) \rfloor$, and $\underlineκ < \overlineκ$ the solutions of $κ(1-\log(κ)) = ω$. Since the error of approximating $li(x)$ using Stieltjes asymptotic series $\displaystyle li_{*}(x) = \frac{x}{\log(x)}\sum_{k=0}^{n-1}\frac{k!}{\log^{k}(x)} + (\log(x)-n)\frac{xn!}{\log^{n+1}(x)}$, with $\displaystyle n = \lfloor \log(x) \rfloor$ for all $x\ge e$, satisfies $\displaystyle |\varepsilon(x)| = |li(x)-li_{*}(x)| \le 1.265692883422\ldots$, by using Stirling's approximation and some facts about $\log(x)$ and floor functions, we show that $\displaystyle \varepsilon_{0}(x) = li(x) - li_{\underline{1/2},0}(x)$, $\displaystyle \underline{\varepsilon}(x) = li(x) - li_{\underline{1/2},\underlineκ\log(x)}(x)$, $\displaystyle \overline{\varepsilon}(x) = li(x) - li_{\overline{1/2},\overlineκ\log(x)}(x)$, and $\varepsilon_{1}(x) = li_{\overline{1/2},1}(x) - li(x)$ belong to $O\left(\sqrt{\frac{x}{\log(x)}}\right)$. Moreover, we conjecture that $li_{0}(x) \le π(x) \le li_{1}(x)$ and $\underline{li}(x) \le π(x) \le \overline{li}(x)$ for all $x \ge e$, here $π(x)$ is the prime counting function and we show that if one of those conjectures is true then the Riemann Hypothesis is true.