Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  355.10034
Autor:  Erdös, Paul; Richards, Ian
Title:  Density functions for prime and relatively prime numbers. (In English)
Source:  Monatsh. Math. 83, 99-112 (1977).
Review:  This is a sequel to papers by P.Erdös and J.L.Selfridge [Proc. Manitoba Conf. numer. Math. 1971, 1-14 (1971; Zbl 267.10054)] and D.Hensley and I.Richards [Acta Arith. 25, 375-391 (1974; Zbl 285.10004)]. Let \rho^*(x) be the maximum number of primes in any interval beyond x of length x. Let r^*(x) be the maximum number of pairwise coprime integers in any interval of length x. A finite set S of integers is ''\rho^*-admissible'' if for each prime p some residue class (mod p) excludes all elements of S. S is ''r^*-admissible'' if for each prime p some residue class (mod p) excludes all but at most one element of S. The prime k-tuples hypothesis asserts that if {b1 < b2 < ... < bk} is \rho^*-admissible then there are infinitely many positive integers n for which all of n+b1, n+b2, ..., n+bk are prime. Under the prime k-tuples hypothesis it is proved that \rho^*(x) is the number of elements in a maximal \rho^*-admissible set in any interval of length x (proposition 4). With no hypothesis (proposition 5) r^*(x) is the maximum number of elements in any r^*-admissible set in any interval of length x.
Sieve methods are used to get upper an lower bounds on r^*(x)-\rho^*(x). Namely, theorem 1: There is an effectively computable c > 0 for which r^*(x)-\rho^*(x) > xc for all sufficiently large x. Theorem 2: Under the prime k-tuples hypothesis,

r^*(x)-\rho^*(x) = o(x/ log2x) as  x ––> oo.

The previously known lower bound was log x. Since Hensley and Richards have proved under the prime k-tuples hypothesis that \rho^*(x) < \pi(x)+Kx/ log2x, then it appears that r^*(x) ~ \rho^*(x). This is not surprising, however, under the prime k-tuples hypothesis we have the even stronger fact that r^*(X)-\rho^*(X) = o(x/ log2x) where as \rho^*(x)-\pi(x) > Kx/ log2x. Thus it seems that \rho^*(x) is much closer to r^*(x) than to \pi(x). Of course, the prime k-tuples hypothesis is a rather strong assumption which has as yet not been verified even for k = 2.
Reviewer:  J.P.Tull
Classif.:  * 11N05 Distribution of primes
                   11N35 Sieves

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