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On arithmetic functions related to consecutive divisors. (Sur les fonctions arithmétiques liées aux diviseurs consécutifs.) (French) Zbl 0676.10030

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 \(1=d_1<d_2<...<d_{\tau(n)}=n\) denote the divisors of \(n\); the authors study, amongst others, the functions \[ f(n)= \text{card}\{i:\quad 1\leq i<\tau(n),\quad (d_i,d_{i+1})=1\},\quad H(n)=\sum_{1\leq i<\tau(n)}(d_{i+1}-d_i)^{-1}. \] 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 \(\max_{n\leq x} f(n)\), 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 \(\sum_{n\leq x}f(n)\) in [Bull. Soc. Math. Fr. 111, 125-145 (1983; Zbl 0526.10036)] and the error term in the formula for \(\sum_{n\leq x}H(n)\) established by A. Ivić and J.-M. De Koninck in [Can. Math. Bull. 29, 208-217 (1986; Zbl 0543.10034)].
Reviewer: E.J.Scourfield

MSC:

11N05 Distribution of primes
11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions in probabilistic number theory
11B83 Special sequences and polynomials
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References:

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