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dc.contributor.authorKolossváry, István B.
dc.contributor.authorKolossváry, István T.
dc.date.accessioned2023-01-25T00:38:16Z
dc.date.available2023-01-25T00:38:16Z
dc.date.issued2022-05
dc.identifier.citationKolossváry , I B & Kolossváry , I T 2022 , ' Distance between natural numbers based on their prime signature ' , Journal of Number Theory , vol. 234 , pp. 120-139 . https://doi.org/10.1016/j.jnt.2021.09.010en
dc.identifier.issn0022-314X
dc.identifier.otherPURE: 271862390
dc.identifier.otherPURE UUID: f7140d5f-3938-445e-bab0-8b6e820a41f0
dc.identifier.otherArXiv: http://arxiv.org/abs/2005.02027v2
dc.identifier.otherScopus: 85118341682
dc.identifier.otherWOS: 000795908300006
dc.identifier.urihttp://hdl.handle.net/10023/26824
dc.descriptionFunding: ITK was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034).en
dc.description.abstractEach natural number is uniquely determined by its prime signature, an infinite dimensional vector indexed by the prime numbers in increasing order. We use this to define a new metric between natural numbers induced by the l∞ norm of the signatures. In this space, we look at the natural analog of the number line and, in particular, study the arithmetic function L∞(N), which tabulates the cumulative sum of distances between consecutive natural numbers up to N in this new metric.  Our main result is to identify the positive and finite limit of the sequence L∞(N)/N as the expectation of a certain random variable. The main technical contribution is to show with elementary probability that for K=1, 2 or 3 and ω0,= ..., = ωK ≥ 2 the following asymptotic density holds lim n→ ∞ |{M ≤ n: ||M-j||∞ < ωj for j = 0, ..., K}|/n= π p:prime (1-KΣj=0 1/pωj) This is a generalization of the formula for k-free numbers, i.e. when ω0 = ... = ωK = k. The random variable is derived from the joint distribution when K = 1. As an application, we obtained a modified version of the prime number theorem. our computations up to N = 1012 have also revealed that prime gaps show a considerably richer structure than on the traditional number line. Moreover, we raise additional open problems, which could be of independent interest.
dc.language.isoeng
dc.relation.ispartofJournal of Number Theoryen
dc.rightsCopyright © 2021 Elsevier Inc. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jnt.2021.09.010.en
dc.subjectPower-free numbersen
dc.subjectPrime griden
dc.subjectLimiting densitiesen
dc.subjectDistribution of primesen
dc.subjectQA Mathematicsen
dc.subjectT-NDASen
dc.subject.lccQAen
dc.titleDistance between natural numbers based on their prime signatureen
dc.typeJournal articleen
dc.contributor.sponsorThe Leverhulme Trusten
dc.description.versionPostprinten
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.identifier.doihttps://doi.org/10.1016/j.jnt.2021.09.010
dc.description.statusPeer revieweden
dc.identifier.urlhttps://arxiv.org/abs/2005.02027en
dc.identifier.grantnumberRPG-2019-034en


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