论文标题
布朗弱极限，对于整数的随机m培养
A Brownian weak limit for the least common multiple of a random m-tuple of integers
论文作者
论文摘要
令$ b_n（m）$为$ \ {1,2，\ ldots，n \} $的所有$ m $ elements子集中的均匀选择集。我们提供集合的路线构造$（b_n（m））_ {1 \ leq m \ leq n} $，并证明$（b_n（\ lfloor mt \ rfloor）中，整数最不常见的倍数的对数_ {无穷大。我们的方法包括两个步骤。首先，我们表明上述结果是$ \ {1,2，\ ldots，n \} $的$ m $独立随机变量的对数的多维中心限制定理的结果。其次，我们通过样品的质量分解中被忽略的多重性来提供随机样品中最不常见的倍数的新颖近似值。
Let $B_n(m)$ be a set picked uniformly at random among all $m$-elements subsets of $\{1,2,\ldots,n\}$. We provide a pathwise construction of the collection $(B_n(m))_{1\leq m\leq n}$ and prove that the logarithm of the least common multiple of the integers in $(B_n(\lfloor mt\rfloor))_{t\geq 0}$, properly centered and normalized, converges to a Brownian motion when both $m,n$ tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of $m$ independent random variables having uniform distribution on $\{1,2,\ldots,n\}$. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.
