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**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 **{**b_{1} < b_{2} < ... < b_{k}**}** is \rho^*-admissible then there are infinitely many positive integers n for which all of n+b_{1}, n+b_{2}, ..., n+b_{k} 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) > x^{c} for all sufficiently large x. Theorem 2: Under the prime k-tuples hypothesis, r^*(x)-\rho^*(x) = o(x/ log^{2}x) 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/ log^{2}x, 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/ log^{2}x) where as \rho^*(x)-\pi(x) > Kx/ log^{2}x. 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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