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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.identifier271862390
dc.identifierf7140d5f-3938-445e-bab0-8b6e820a41f0
dc.identifier85118341682
dc.identifier000795908300006
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.otherArXiv: http://arxiv.org/abs/2005.02027v2
dc.identifier.otherORCID: /0000-0002-2216-305X/work/133736960
dc.identifier.urihttps://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.format.extent1910007
dc.language.isoeng
dc.relation.ispartofJournal of Number Theoryen
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.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.identifier.doihttps://doi.org/10.1016/j.jnt.2021.09.010
dc.description.statusPeer revieweden
dc.date.embargoedUntil2023-01-25
dc.identifier.urlhttps://arxiv.org/abs/2005.02027en
dc.identifier.grantnumberRPG-2019-034en


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