论文标题
二项式系数的产品的部分因素化
Partial Factorizations of Products of Binomial Coefficients
论文作者
论文摘要
令$ g_n = \ prod_ {k = 0}^n \ binom {n} {k}，$ $ n $ th $ n $ th ROW PASCAL三角形的元素的乘积。本文研究了所有主要因素的$ g（n，x）$ p $ $ g_n $具有$ p \ le x $的$ g（n，x）$的$ g_n $的部分因素。它显示$ \ log g（n，αn）\ sim f_g（α）n^2 $ as $ n \ to \ infty $用于限制函数$ f_ {g}（α）$以$ 0 \ leleα\ le 1 $定义。主要结果是根据功能的研究$ a（n，x），b（n，x），编码整数$ n $（和较小的整数）的基本$ p $ radix扩展的统计数据的$，其中基本$ p $远比Primes $ p $ $ p \ p \ le x $。 $ a（n，x）$和$ b（n，x）$的渐近学使用带有剩余期限或有条件的riemann假设中的素数定理得出。
Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the $n$-th row of Pascal's triangle. This paper studies the partial factorizations of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having $p \le x$, counted with multiplicity. It shows $\log G(n, αn) \sim f_G(α)n^2$ as $n \to \infty$ for a limit function $f_{G}(α)$ defined for $0 \le α\le 1$. The main results are deduced from study of functions $A(n, x), B(n,x),$ that encode statistics of the base $p$ radix expansions of the integer $n$ (and smaller integers), where the base $p$ ranges over primes $p \le x$. Asymptotics of $A(n,x)$ and $B(n,x)$ are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.
