Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  683.10035
Autor:  Erdös, Paul; Nicolas, Jean-Louis
Title:  Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier. (Large values of functions connected to consecutive prime divisors of an integer.) (In French)
Source:  Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 169-200 (1989).
Review:  [For the entire collection see Zbl 674.00008.]
Let n = q1\alpha1... qk\alphak be the standard factorization of n into primes. The authors are interested in large values of the two functions

f(n) = sumk-1i = 1qi/qi+1,  F(n) = sumk-1i = 1(1-qi/qi+1).

The main results are as follows: (i) There exists a constant C > 0 such that, as n ––> oo, F(n) \leq \sqrt{log n}-C+o(1), with equality holding for infinitely many n. (ii) Call an integer N > 1 an f-champion if f(N) > f(n) for every n < N. Then, for every sufficiently large k, the number Nk = p1... pk, where pi denotes the ith prime, is an f-champion. Moreover, under the assumption of Crámer's conjecture pi+1-pi << (log pi)2, every sufficiently large f- champion is of the form Nk or Nk/p for some prime factor p of Nk.
Reviewer:  A.Hildebrand
Classif.:  * 11N05 Distribution of primes
                   11N37 Asymptotic results on arithmetic functions
Keywords:  arithmetic function; prime divisors; maximal order of magnitude; large values; f-champion
Citations:  Zbl 674.00008

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