On the least common multiple of a set of integers not exceeding N
On the least common multiple of a set of integers not exceeding N
复制标题
DOI：
10.1016/1385-7258(80)90019-0
复制
发表时间：
1980
期刊：
影响因子：
--
通讯作者：
C. J. Goutziers
中科院分区：
文献类型：
--
作者：
C. J. Goutziers
设c是一个真实的数0-CC< 1。假设X%是不超过N的任意[cN]个正整数的集合。关于B（X $）可以说些什么？本文证明了：若EP-0，则当N充分大时，log CJ（X $）>（1 E）~ ZP.这里p和q是依赖于c的常数。这个下界是最好的，因为另一方面我们证明了，对于N非常大，存在一个不超过N的正整数集合AGN，其性质为：log a（&）<（1+ e）pNQ，其中p和q是与下界相同的常数。我们注意到，由于[I，N]中素数的个数小于n [CNj]，所以存在一个[CNj]的正整数集合B $，G（Ev）= I.因此，平凡上界G（X $）< C（KN）不能被改进。L
Let c be a real number 0-CC< 1. Suppose X% is any set of [cN] positive integers not exceeding N. What can be said about B (X $)? We prove; If EP-0 we have, log CJ (X $)>(lE)~ ZP, for N sufficiently large. Here p and q are constants depending on c. This lowerbound is the best possible since on the other hand we show that, for N sticiently large, there exists a set AGN of [cNj positive integers not exceeding N with the property: log a (&)<(1+ e) pNQ, where p and q are the same constants as in the lowerbound. We remark that, since the number of primes in [I, N] is smaller tha, n [cNj, there exists a set B $ of [ciVj positive integers with G (Ev)= I. Therefore the trivial upperbound G (X $)< C (KN) can not be improved. l