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

Abstract

We introduce approximation functions of $li(x)$ for all $x\ge e$: (1) $l{i}_{\underline{\omega },\alpha }(x)=\frac{x}{\log (x)}\left(\alpha \frac{\underline{m}!}{{\log }^{\underline{m}}(x)}+\sum _{k=0}^{\underline{m}-1}\frac{k!}{{\log }^k(x)}\right)$, and (2) $l{i}_{\overline{\omega },\beta }=\frac{x}{\log (x)}\left(\beta \frac{\overline{m}!}{{\log }^{\overline{m}}(x)}+\sum _{k=0}^{\overline{m}-1}\frac{k!}{{\log }^k(x)}\right)$ with $0<\omega <1$ a real number, $\alpha \in \{0,\underline{\kappa }\log (x)\}$, $\underline{m}=\lfloor \underline{\kappa }\log (x)\rfloor$, $\beta \in \{\overline{\kappa }\log (x),1\}$, $\overline{m}=\lfloor \overline{\kappa }\log (x)\rfloor$, and $\underline{\kappa }<\overline{\kappa }$ the solutions of $\kappa (1-\log (\kappa ))=\omega$. Since the error of approximating $li(x)$ using Stieltjes asymptotic series $l{i}_{\ast }(x)=\frac{x}{\log (x)}\sum _{k=0}^{n-1}\frac{k!}{{\log }^k(x)}+(\log (x)-n)\frac{xn!}{{\log }^{n+1}(x)}$, with $n=\lfloor \log (x)\rfloor$ for all $x\ge e$, satisfies $|\epsilon (x)|=|li(x)-l{i}_{\ast }(x)|\le 1.265692883422\dots$, by using Stirling's approximation and some facts about $\log (x)$ and floor functions, we show that ${\epsilon }_0(x)=li(x)-l{i}_{\underline{1/2},0}(x)$, $\underline{\epsilon }(x)=li(x)-l{i}_{\underline{1/2},\underline{\kappa }\log (x)}(x)$, $\overline{\epsilon }(x)=li(x)-l{i}_{\overline{1/2},\overline{\kappa }\log (x)}(x)$, and ${\epsilon }_1(x)=l{i}_{\overline{1/2},1}(x)-li(x)$ belong to $O\left(\sqrt{\frac{x}{\log (x)}}\right)$. Moreover, we conjecture that $l{i}_0(x)\le \pi (x)\le l{i}_1(x)$ and $\underline{li}(x)\le \pi (x)\le \overline{li}(x)$ for all $x\ge e$, here $\pi (x)$ is the prime counting function and we show that if one of those conjectures is true then the Riemann Hypothesis is true.

17 pages, 1 Figure, 3 Tables