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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 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 language | English |
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Pages (from-to) | 120-139 |
Journal | Journal of Number Theory |
Volume | 234 |
Early online date | 21 Oct 2021 |
DOIs | |
Publication status | Published - May 2022 |
Keywords
- Power-free 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
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New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
1/09/19 → 31/01/23
Project: Standard