Distance between natural numbers based on their prime signature

Abstract

We define a new metric between natural numbers induced by the ${\ell }_{\infty }$ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function $L_{\infty }(N)$, 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 $L_{\infty }(N)/N$ 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 ${\omega }_0,\dots ,{\omega }_K\ge 2$ the following asymptotic density holds $$\underset{n\to \infty }{\lim }\frac{\left|\right\{M\le n:\Vert M-j{\Vert }_{\infty }<{\omega }_j\text{ for }j=0,\dots ,K\left\}\right|}{n}=\prod _{p:\mathrm{prime}}(1-\sum _{j=0}^K\frac{1}{p^{{\omega }_j}}).$$ This is a generalization of the formula for $k$-free numbers, i.e. when ${\omega }_0=\dots ={\omega }_K=k$. 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 $N=10^{12}$ 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.

This article supersedes arXiv:1711.02903. v3: accepted version in J. Number Theory with additional Appendix. v2: Conjecture 1 and 2 of v1 are now proved, major revision of exposition

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