论文标题
自然数量基于其主要签名之间的距离
Distance between natural numbers based on their prime signature
论文作者
论文摘要
我们定义了由$ \ ell_ \ infty $ norm在其独特的Prime签名中引起的自然数量之间的新指标。在这个空间中，我们查看数字线的自然类似物，并研究算术函数$ l_ \ infty（n）$，该$列出了该新指标中连续的自然数量高达$ n $之间的距离累计总和。 我们的主要结果是确定序列$ l_ \ infty（n）/n $的正和有限限制为某个随机变量的期望。主要的技术贡献是要以$ k = 1,2 $或$ 3 $和$ω_0，\ ldots，ω_k\ geq 2 $以下渐近密度容纳$ \ lim_ {n \ to \ infty} \ frac { \ | M-J \ | _ \ infty <ω_j\ text {for} j = 0，\ ldots，k \ big \} \ big |} {n} = \ prod_ {p：\，\ \ \ \ \ \ \ \ m mathrm {prime}}}}} \！ \ bigG（1- \ sum_ {j = 0}^k \ frac {1} {p^{ω_j}}} \ bigg）〜。 $$这是对$ k $ free数字的公式的概括，即当$ω_0= \ ldots =ω__k= k $时。 $ k = 1 $时，随机变量是从关节分布得出的。 作为应用程序，我们获得了质数定理的修改版本。我们的计算高达$ n = 10^{12} $还表明，主要差距显示出比传统数字线更丰富的结构。此外，我们提出了其他开放问题，这可能具有独立的兴趣。
We define a new metric between natural numbers induced by the $\ell_\infty$ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function $L_\infty(N)$, which tabulates the cumulative sum of distances between consecutive natural numbers up to $N$ in this new metric. Our main result is to identify the positive and finite limit of the sequence $L_\infty(N)/N$ as the expectation of a certain random variable. The main technical contribution is to show with elementary probability that for $K=1,2$ or $3$ and $ω_0,\ldots,ω_K\geq 2$ the following asymptotic density holds $$ \lim_{n\to\infty}\frac{\big|\big\{M\leq n:\; \|M-j\|_\infty <ω_j \text{ for } j=0,\ldots,K \big\}\big|}{n} = \prod_{p:\, \mathrm{prime}}\! \bigg( 1- \sum_{j=0}^K\frac{1}{p^{ω_j}} \bigg)~. $$ This is a generalization of the formula for $k$-free numbers, i.e. when $ω_0=\ldots=ω_K=k$. 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.
