## 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 = q1\alpha1q2\alpha2... qk\alphak, where q1 < q2 < ... < qk are primes. In this paper four arithmetical functions are considered, namely

f(n) = sumk-1i = 1\frac{qi}{qi+1};   F(n) = sumk-1i = 1(1-\frac{qi}{qi+1});

h(n) = sumk-1i = 1\frac{1}{qi+1-qi};   \hat h(n) = sum1 \leq i < j \leq k\frac{1}{qj-qi}.

Previously, the authors had shown that n(x) = prodp \leq xp, 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+x1/100.