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 holdslimn→∞⁡|{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.