An effective Bombieri-Vinogradov error term for sifting problems

[2510.10853] An effective Bombieri-Vinogradov error term for sifting problems Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2510.10853 (math) [Submitted on 12 Oct 2025 ( v1 ), last revised 25 Aug 2026 (this version, v3)] Title: An effective Bombieri-Vinogradov error term for sifting problems Authors: Daniel R. Johnston View PDF HTML (experimental) Abstract: In number theory, many major results related to the additive properties of primes are proven using the methods of sieve theory. However, in nearly every case, the existing proofs of these results are ineffective, in that explicit values for which they hold cannot be computed. The reason for this ineffectivity is due to the reliance on the Bombieri--Vinogradov theorem. In this paper, we show that any classical sifting problem with a Bombieri--Vinogradov style error term can in fact be made effective, with no loss to the asymptotic form of the original (ineffective) result. This is done by carefully modifying the sieve upper and lower bounds as to avoid the usual complications regarding the existence of a Siegel zero. We also provide some simple applications. For example, we show that one may effectively bound the number of primes $p\leq x$ such that $p+2$ is also prime by \begin{equation*}(4+o(1))C_2\frac{x}{(\log x)^2},\end{equation*}where\begin{equation*}C_2=2\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right).\end{equation*} Comments: 32 pages, to appear in Mathematika Subjects: Number Theory (math.NT) MSC classes: 11N35 (Primary) 11M20, 11N36 (Secondary) Report number: MPIM-Bonn-2026 Cite as: arXiv:2510.10853 [math.NT] (or arXiv:2510.10853v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.2510.10853 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Daniel R. Johnston [ view email ] [v1] Sun, 12 Oct 2025 23:33:01 UTC (32 KB) [v2] Sun, 5 Apr 2026 02:03:48 UTC (27 KB) [v3] Tue, 25 Aug 2026 08:00:23 UTC (28 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-10 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation 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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