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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 687.10031

**Autor: ** Erdös, Paul; Nicolas, J.L.

**Title: ** On functions connected with prime divisors of an integer. (In English)

**Source: ** Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 381-391 (1989).

**Review: ** [For the entire collection see Zbl 676.00005.]

For an arithmetical function g an integer n is called a g-champion if g(m) < g(n) for all m < n. Let n = q_{1}^{\alpha1}q_{2}^{\alpha2}... q_{k}^{\alphak}, where q_{1} < q_{2} < ... < q_{k} are primes. In this paper four arithmetical functions are considered, namely f(n) = **sum**^{k-1}_{i = 1}\frac{q_{i}}{q_{i+1}}; F(n) = **sum**^{k-1}_{i = 1}(1-\frac{q_{i}}{q_{i+1}});

h(n) = **sum**^{k-1}_{i = 1}\frac{1}{q_{i+1}-q_{i}}; \hat h(n) = **sum**_{1 \leq i < j \leq k}\frac{1}{q_{j}-q_{i}}. Previously, the authors had shown that n(x) = **prod**_{p \leq x}p, p = prime, is an f-champion for all large x, but not an F-champion for large x [Théorie des nombres, C. R. Conf. Int., Quebec/Can. 1987, 169- 200 (1989; Zbl 683.10035)]. Here the authors first show that n(x) is not an h-champion, by using a deep result due to Maier on chains of long gaps between primes. Next, they show that n(x) is not an \hat h-champion by assuming two strong conjectures one of which is that in short intervals of the form (x,x+x^{\epsilon}) one can find the expected number of primes. Their second assumption is that there exist 4-tuples of primes p,p+2,p+6,p+8 in short intervals of the form x,x+x^{1/100}.

**Reviewer: ** K.Alladi

**Classif.: ** * 11N37 Asymptotic results on arithmetic functions

11N05 Distribution of primes

**Keywords: ** champion numbers; primes in short intervals; gaps between primes

**Citations: ** Zbl 676.00005; Zbl 683.10035

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