Density functions for prime and relatively prime numbers

Abstract

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)<x ^c for somec>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.