Papers › On the Stieltjes Approximation Error to Logarithmic Integral
On the Stieltjes Approximation Error to Logarithmic Integral
Jonatan Gomez
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We study the approximation error ε(x)=li_*(x)-li(x) arising from the classical Stieltjes asymptotic expansion for the logarithmic integral. Our analysis is based on the discrete values εₖ=ε(eᵏ) and their increments Δₖ=εₖ₊₁-εₖ, for which we derive new unconditional analytic bounds. Using precise integral representations for each increment Δₖ, together with sharp upper and lower estimates for the associated kernel integrals, we obtain computable and uniform bounds for εₖ for all k≥1, and hence for ε(x) for all x≥e. We prove the following unconditional bounds: [ 1/3√(2π/(ln(x))) + o(1/(√(ln(x)))) ≤ε(x) ≤1/3√(2π/(ln(x))) + o(1/(√(ln(x)))) for all e ≤x ≤e¹⁰⁰⁰, ] [ 1/3√(2π/(ln(x))) + o(1/(√(ln(x)))) - Cₗ ≤ε(x) ≤1/3√(2π/(ln(x))) + o(1/(√(ln(x)))) + Cᵣ for all x>e¹⁰⁰⁰ with Cₗ = 0.0000035462 and Cᵣ=0.0000021511. ] These results establish the first fully explicit global bounds for the Stieltjes approximation error. Finally, our findings strongly support the conjectural behaviour: ε(x) = 1/3√(2π/(ln(x))) + o(1/(√(ln(x)))), x≥e.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections