{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-stieltjes-approximation-error-to","title":"On the Stieltjes Approximation Error to Logarithmic Integral","arxiv_id":"2406.12152","date":"2024-06-17","proceeding":null,"authors":["Jonatan Gomez"],"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: $$ \\varepsilon(x) = \\frac{1}{3}\\sqrt{\\frac{2\\pi}{\\ln(x)}} + o\\!\\left(\\frac{1}{\\sqrt{\\ln(x)}}\\right), \\qquad x\\ge e. $$","url_abs":"https://arxiv.org/abs/2406.12152v1","url_pdf":"https://arxiv.org/pdf/2406.12152v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-stieltjes-approximation-error-to","repo_url":"https://github.com/jgomezpe/primenumbers","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}