On the Stieltjes Approximation Error to Logarithmic Integral

Abstract

We study the approximation error $\epsilon (x)={\mathrm{li}}_{\ast }(x)-\mathrm{li}(x)$ arising from the classical Stieltjes asymptotic expansion for the logarithmic integral. Our analysis is based on the discrete values ${\epsilon }_k=\epsilon (e^k)$ and their increments ${\Delta }_k={\epsilon }_{k+1}-{\epsilon }_k,$ for which we derive new unconditional analytic bounds. Using precise integral representations for each increment ${\Delta }_k$, together with sharp upper and lower estimates for the associated kernel integrals, we obtain computable and uniform bounds for ${\epsilon }_k$ for all $k\ge 1$, and hence for $\epsilon (x)$ for all $x\ge e$. We prove the following unconditional bounds: $$\begin{array}{l}\frac{1}{3}\sqrt{\frac{2\pi }{\ln (x)}}+o\left(\frac{1}{\sqrt{\ln (x)}}\right)\le \epsilon (x)\le \frac{1}{3}\sqrt{\frac{2\pi }{\ln (x)}}+o\left(\frac{1}{\sqrt{\ln (x)}}\right)\text{for all }e\le x\le e^{1000},\end{array}$$ $$\begin{array}{l}\frac{1}{3}\sqrt{\frac{2\pi }{\ln (x)}}+o\left(\frac{1}{\sqrt{\ln (x)}}\right)-C_l\le \epsilon (x)\le \frac{1}{3}\sqrt{\frac{2\pi }{\ln (x)}}+o\left(\frac{1}{\sqrt{\ln (x)}}\right)+C_r\text{for all }x>e^{1000}\text{ with }{C}_l=0.0000035462\text{ and }{C}_r=\mathrm{0.0000021511.}\end{array}$$ These results establish the first fully explicit global bounds for the Stieltjes approximation error. Finally, our findings strongly support the conjectural behaviour: $$\epsilon (x)=\frac{1}{3}\sqrt{\frac{2\pi }{\ln (x)}}+o\left(\frac{1}{\sqrt{\ln (x)}}\right),x\ge e.$$

26 ppages