Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  676.10030
Autor:  Erdös, Paul; Tenenbaum, G.
Title:  Sur les fonctions arithmétiques liées aux diviseurs consécutifs. (On arithmetic functions related to consecutive divisors.) (In French)
Source:  J. Number Theory 31, No.3, 285-311 (1989).
Review:  There is now a wealth of literature on problems concerning consecutive divisors of an integer, to which the present paper makes a further interesting contribution. Let 1 = d1 < d2 < ... < d\tau(n) = n denote the divisors of n; the authors study, amongst others, the functions

f(n) = card{i:   1 \leq i < \tau(n),  (di,di+1) = 1},  H(n) = sum1 \leq i < \tau(n)(di+1-di)-1.

The results obtained are too complicated and numerous to state here, but we indicate the type of problems investigated. The authors derive, for example, estimates from above for f(n) and from below for maxn \leq x f(n), with a similar treatment for H(n), and they show that H has a distribution function. They are able to improve their own upper estimate for sumn \leq xf(n) in [Bull. Soc. Math. Fr. 111, 125-145 (1983; Zbl 526.10036)] and the error term in the formula for sumn \leq xH(n) established by A. Ivic and J.-M. De Koninck in [Can. Math. Bull. 29, 208-217 (1986; Zbl 543.10034)].
Reviewer:  E.J.Scourfield
Classif.:  * 11N05 Distribution of primes
                   11N37 Asymptotic results on arithmetic functions
                   11K65 Arithmetic functions (probabilistic number theory)
                   11B83 Special sequences of integers and polynomials
Keywords:  consecutive divisors; estimates from above; distribution function; upper estimate; error term
Citations:  Zbl 585.10030; Zbl 526.10036; Zbl 543.10034

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