On the Stieltjes Approximation Error to Logarithmic Integral

We study the approximation error $\varepsilon(x)=\operatorname{li}_{*}(x)-\operatorname{li}(x)$ arising from the classical Stieltjes asymptotic expansion for the logarithmic integral. Our analysis is based on the discrete values $\varepsilon_k=\varepsilon(e^{k})$ and their increments $\Delta_k=\varepsilon_{k+1}-\varepsilon_k,$ 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 $\varepsilon_k$ for all $k\ge 1$, and hence for $\varepsilon(x)$ for all $x\ge e$. We prove the following unconditional bounds: $$\begin{array}{l} \displaystyle \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) \le \varepsilon(x) \le \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) \text{for all } e \le x \le e^{1000}, \end{array}$$ $$\begin{array}{l} \displaystyle \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) - C_{l} \le \varepsilon(x) \le \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) + C_{r} \text{for all } x>e^{1000} \text{ with } C_{l} = 0.0000035462\text{ and } C_{r}=0.0000021511. \end{array}$$ These results establish the first fully explicit global bounds for the Stieltjes approximation error. Finally, our findings strongly support the conjectural behaviour: $$\varepsilon(x) = \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\!\left(\frac{1}{\sqrt{\ln(x)}}\right), \qquad x\ge e.$$