On the Stieltjes Approximation Error to Logarithmic Integral
Jonatan Gomez
TL;DR
The paper investigates the Stieltjes approximation error $\varepsilon(x)=\operatorname{li}_{*}(x)-\operatorname{li}(x)$ for the logarithmic integral, linking it to the discrete sequence $\varepsilon_k=\varepsilon(e^{k})$ and increments $\Delta_k$. It develops unconditional bounds and explicit integral representations to control both the continuous error and its discrete analogs, ultimately providing fully explicit global bounds for $\varepsilon(x)$ on $[e,\infty)$ and compelling numerical evidence for the conjectured asymptotics $\varepsilon(x)=\frac{1}{3}\sqrt{\frac{2\pi}{\ln x}}+o(\frac{1}{\sqrt{\ln x}})$. The results rely on a structural decomposition of $\varepsilon(x)$, monotonicity, and asymptotics for the $k$-th error $\varepsilon_k$ and the partial error $\Delta_k$, supported by unimodality arguments and Riemann-sum bounds. Collectively, these findings provide rigorous, computable error controls for Stieltjes-type expansions of the logarithmic integral, with implications for prime-counting approximations and analytic number theory. The work also offers detailed, explicit constants and bounds, strengthening the practical applicability of the Stieltjes expansion in numerical computations.
Abstract
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 $Δ_k=\varepsilon_{k+1}-\varepsilon_k,$ for which we derive new unconditional analytic bounds. Using precise integral representations for each increment $Δ_k$, 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: $$\begin{array}{l} \displaystyle \frac{1}{3}\sqrt{\frac{2π}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) \le \varepsilon(x) \le \frac{1}{3}\sqrt{\frac{2π}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) \text{for all } e \le x \le e^{1000}, \end{array} $$ $$\begin{array}{l} \displaystyle \frac{1}{3}\sqrt{\frac{2π}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) - C_{l} \le \varepsilon(x) \le \frac{1}{3}\sqrt{\frac{2π}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) + C_{r} \text{for all } x>e^{1000} \text{ with } C_{l} = 0.0000035462\text{ and } C_{r}=0.0000021511. \end{array}$$ These results establish the first fully explicit global bounds for the Stieltjes approximation error. Finally, our findings strongly support the conjectural behaviour: $$ \varepsilon(x) = \frac{1}{3}\sqrt{\frac{2π}{\ln(x)}} + o\!\left(\frac{1}{\sqrt{\ln(x)}}\right), \qquad x\ge e. $$