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Distance between natural numbers based on their prime signature

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Date
05/2022
Author
Kolossváry, István B.
Kolossváry, István T.
Funder
The Leverhulme Trust
Grant ID
RPG-2019-034
Keywords
Power-free numbers
Prime grid
Limiting densities
Distribution of primes
QA Mathematics
T-NDAS
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Abstract
Each 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.
Citation
Kolossváry , 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.2021.09.010
Publication
Journal of Number Theory
Status
Peer reviewed
DOI
https://doi.org/10.1016/j.jnt.2021.09.010
ISSN
0022-314X
Type
Journal article
Rights
Copyright © 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.
Description
Funding: ITK was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034).
Collections
  • University of St Andrews Research
URL
https://arxiv.org/abs/2005.02027
URI
http://hdl.handle.net/10023/26824

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