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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 kfree 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 language  English 

Pages (fromto)  120139 
Journal  Journal of Number Theory 
Volume  234 
Early online date  21 Oct 2021 
DOIs  
Publication status  Published  May 2022 
Keywords
 Powerfree numbers
 Prime grid
 Limiting densities
 Distribution of primes
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Dive into the research topics of 'Distance between natural numbers based on their prime signature'. Together they form a unique fingerprint.Projects
 1 Finished

New perspectives in the dimension: New perspectives in the dimension theory of fractals
1/09/19 → 31/01/23
Project: Standard