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 $\omega_0,\ldots,\omega_K\geq 2$ the following asymptotic density holds $$ \lim_{n\to\infty}\frac{\big|\big{M\leq n:\; |M-j|\infty <\omega_j \text{ for } j=0,\ldots,K \big}\big|}{n} = \prod{p:\, \mathrm{prime}}! \bigg( 1- \sum_{j=0}^{K\frac{1}{p{\omega_j}}} \bigg)~. $$ This is a generalization of the formula for $k$-free numbers, i.e. when $\omega_0=\ldots=\omega_K=k$. 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.