On the Product of the Primes
On the Product of the Primes
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DOI：
10.4153/cmb-1972-007-7
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发表时间：
1972-03
期刊：
影响因子：
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通讯作者：
D. Hanson
中科院分区：
文献类型：
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作者：
D. Hanson
近年来，人们多次尝试对小于等于给定整数\(n\)的素数之积进行估计。用\(\pi(n)\)表示上述素数之积，并按通常方式定义。对二项式系数和多项式系数的分析得出了诸如\(A(n)<4^n\)这样的结果，这是由埃尔德什（Erdős）和卡尔马（Kalmár）得出的（见[2]）。莫泽（Moser）的一篇笔记[3]给出了\(A(n)<(3.37)^n\)的一个归纳证明，塞尔弗里奇（Selfridge，未发表）证明了\(A(n)<(3.05)^n\)。人们还知道一些更精确的结果，特别是在罗瑟（Rosser）和舍恩菲尔德（Schoenfeld）的一篇论文[4]中，他们证明了\(\Theta(n)<1.01624^n\)；然而，他们的方法要深奥得多，涉及复变函数理论以及大量的计算。我们仅用初等方法将证明以下定理，该定理在很大程度上改进了[2]和[3]中的结果。
In recent years several attempts have been made to obtain estimates for the product of the primes less than or equal to a given integer n. Denote by the above-mentioned product and define as usual Analysis of binomial and multinomial coefficients has led to results such as A(n)<4n, due to Erdôs and Kalmar (see [2]). A note by Moser [3] gave an inductive proof of A(n)<(3.37)n, and Selfridge (unpublished) proved A(n)<(3.05)n More accurate results are known, in particular those in a paper of Rosser and Schoenfeld [4] in which they prove Θ(n)< 1.01624n; however their methods are considerably deeper and involve the theory of a complex variable as well as heavy computations. Using only elementary methods we will prove the following theorem, which improves the results of [2] and [3] considerably.