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: $$ \varepsilon(x) = \frac{ 1}{ 3}\sqrt{ \frac{ 2\pi}{ \ln(x)}} + o\!\left(\frac{ 1}{ \sqrt{ \ln(x)}}\right), \qquad x\ge e. $$
Categories: math.NT