An Efficient Algorithm for Estimating Prime Counts

[2606.31761] An Efficient Algorithm for Estimating Prime Counts Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2606.31761 (math) [Submitted on 30 Jun 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: An Efficient Algorithm for Estimating Prime Counts Authors: Artur Samojluk , Artur Siemaszko View a PDF of the paper titled An Efficient Algorithm for Estimating Prime Counts, by Artur Samojluk and Artur Siemaszko View PDF HTML (experimental) Abstract: We develop the Samojluk--Siemaszko (S--S) estimator for the prime-counting function $\pi(x)$ using a non-uniform partition generated by generalized triangular numbers. A cold-start evaluation uses $\big O(\sqrt{x})$ local terms, whereas consecutive partition nodes can be processed with amortized $\big O(1)$ update cost. Updated computations up to $10^{19}$, performed with the correction coefficient $c_T=0.7071$, show accuracy comparable with the Riemann approximation $R(x)$; the two estimators are also asymptotically equivalent at the level of their main term. The correction is written as a one-parameter family $S_{\ell,c}(x)$. Finite-range experiments indicate that effective coefficients lie near $0.7$. We prove asymptotic formulas for the natural scale $q_\ell(x)$ and the accumulated discretization error $D_\ell(x)$, obtaining an unconditional transfer relation between the normalized S--S error and the classical normalized prime-number-theorem error. Together with the logarithmic-mean theorem under RH, this identifies the exact coefficient $c_{\mathrm{th}}=1/\sqrt2$ as uniquely asymptotically optimal in the logarithmic-mean centering sense. The converse implication is quoted from a companion manuscript in preparation. Subjects: Number Theory (math.NT) Cite as: arXiv:2606.31761 [math.NT] (or arXiv:2606.31761v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2606.31761 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Artur Siemaszko [ view email ] [v1] Tue, 30 Jun 2026 14:49:55 UTC (59 KB) [v2] Sat, 15 Aug 2026 11:46:08 UTC (122 KB) Full-text links: Access Paper: View a PDF of the paper titled An Efficient Algorithm for Estimating Prime Counts, by Artur Samojluk and Artur Siemaszko View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-06 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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