Distance between natural numbers based on their prime signature
Abstract
We define a new metric between natural numbers induced by the ℓ∞ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and 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 limn→∞∣∣{M≤n:∥M−j∥∞<ωj for j=0,…,K}∣∣n=∏p:prime(1−∑j=0K1pωj) . 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 obtain 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
ISSN
0022-314XType
Journal article
Description
ITK was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034).Collections
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