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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 ω_{0},= ..., = ω_{K }≥ 2 the following asymptotic density holds
lim n→ ∞ {M ≤ n: Mj_{∞ }< ω_{j }for j = 0, ..., K}/n= π p:prime (1^{K}Σ_{j=0 1/p}ωj)
This is a generalization of the formula for kfree numbers, i.e. when ω_{0} = ... = ω_{K} = 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 = 10^{12 }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.
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 ω_{0},= ..., = ω_{K }≥ 2 the following asymptotic density holds
lim n→ ∞ {M ≤ n: Mj_{∞ }< ω_{j }for j = 0, ..., K}/n= π p:prime (1^{K}Σ_{j=0 1/p}ωj)
This is a generalization of the formula for kfree numbers, i.e. when ω_{0} = ... = ω_{K} = 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 = 10^{12 }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  25 Jan 2022 
DOIs  
Publication status  Published  May 2022 
Keywords
 Powerfree numbers
 Prime grid
 Limiting densities
 Distribution of primes
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 1 Finished

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