On the Stieltjes Approximation Error to Logarithmic Integral

TL;DR

This paper derives explicit, uniform bounds for the approximation error of the logarithmic integral using Stieltjes asymptotics, providing the first global bounds and supporting the conjecture of its asymptotic behavior.

Contribution

It introduces the first fully explicit global bounds for the Stieltjes approximation error to the logarithmic integral, advancing understanding of its asymptotic behavior.

Findings

Established explicit bounds for the approximation error for all x ≥ e.

Provided numerical constants for bounds beyond x > e^{1000}.

Supported the conjecture that the error behaves like (1/3)√(2π/ln x) asymptotically.

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 $\Delta_k=\varepsilon_{k+1}-\varepsilon_k,$ 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: $$ \varepsilon(x) = \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\!\left(\frac{1}{\sqrt{\ln(x)}}\right), \qquad x\ge e. $$