Density functions for prime and relatively prime numbers
Density functions for prime and relatively prime numbers
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素数和互素数的密度函数
DOI：
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发表时间：
1977
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影响因子：
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通讯作者：
I. Richards
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文献类型：
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作者：
P. Erdös;I. Richards
设r *（x）表示长度为x的区间（y，y+x）中两两互素整数的最大个数，设r *（x）表示长度为x的区间（y，y+x）中素数的最大个数，其中y ≥x.在本文中，我们假设“素元元组假设”。(This假设可以通过使用另一种筛理论定义来避免; cf.第1节的开始。）我们研究差分器 *（x）-Σ *（x）：也就是说，我们问在长度为x的区间上可以存在比素数的最大可能数量多多少的相对素数。作为下界，我们得到了r *（x）-∞ *（x）0（当x →∞）.这改进了先前的logx的下限。作为上界，我们得到r *（x）-r *（x）=o[x/（logx）2]。已知π *（x）-π（x）> const。[x/（logx）2];.;因此，与π *（x）-π（x）相比，r *（x）和π *（x）之间的差可以忽略。到目前为止提到的结果涉及“上限”或“最大化”筛。在第2节中，在两种类型的“最小”筛之间进行了类似的比较。其中之一是“擦除”筛子，它完全消除了长度为x的区间;另一个是由埃尔德什和塞尔弗里奇[1]引入的，涉及一种成对相对素数集合的“极大极小”。这两种筛选方法所产生的功能又是密切相关的。
Letr*(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ϱ*(x) denote the maximum number ofprime integers in any interval (y,y+x] wherey≥x. Throughout this paper we assume the “primek-tuples hypothesis.” (This hypothesis could be avoided by using an alternative sievetheoretic definition of ϱ*(x); cf. the beginning of Section 1.) We investigate the differencer*(x)—ϱ*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr*(x)—ϱ*(x)0 (whenx→∞). This improves the previous lower bound of logx. As an upper bound we getr*(x)—ϱ*(x)=o[x/(logx)2]. It is known that ϱ*(x)—π(x)>const.[x/(logx)2];.; thus the difference betweenr*(x) and ϱ*(x) is negligible compared to ϱ*(x)—π(x). The results mentioned so far involve the “upper bound” or “maximizing” sieve. In Section 2, similar comparisons are made between two types of “minimum” sieves. One of these is the “erasing” sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of “minimax” for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.