Distance between natural numbers based on their prime signature

István B. Kolossváry*, István T. Kolossváry*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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→∞∣∣{Mn:∥Mj∥∞<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.

Original languageEnglish
Pages (from-to)120-139
JournalJournal of Number Theory
Volume234
Early online date21 Oct 2021
DOIs
Publication statusPublished - May 2022

Keywords

  • Power-free numbers
  • Prime grid
  • Limiting densities
  • Distribution of primes

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