Distance between natural numbers based on their prime signature

István B. Kolossváry, István T. Kolossváry

Published 2020-05-05Version 1

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 $\ell_\infty$ norm of the signatures. In this space, we look at the natural analog of the number line and, in particular, study the arithmetic function $L_\infty(N)$, which tabulates the accumulative sum of distances between consecutive natural numbers up to $N$ in this new metric. Based on computing $L_\infty(N)$ up to $N=10^{12}$, we found that the ratio $L_\infty(N)/N$ rapidly settles around a specific value $c_0=2.2883695\ldots$. Our main result is to define a random variable, whose expected value seems to agree perfectly with $c_0$. 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 $K=1$. Our computations have also revealed that prime gaps show a considerably richer structure than on the traditional number line. The computations support modified versions of the prime number theorem and Polignac's conjecture, moreover, we raise additional open problems, which could be of independent interest.