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**Zentralblatt MATH**

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

**Zbl.No: ** 718.11041

**Autor: ** Balog, A.; Erdös, Paul; Tenenbaum, G.

**Title: ** On arithmetic functions involving consecutive divisors. (In English)

**Source: ** Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA) 1989, Prog. Math. 85, 77-90 (1990).

**Review: ** [For the entire collection see Zbl 711.00008.]

The primary focus of this paper is the behavior of the functions H(n) = **sum**_{1 \leq i < \tau (n)}(d_{i+1}-d_{i})^{-1}, \kappa(n) = **sum**_{d(d+1)|n}1, where \tau(n) is the divisor function and 1 = d_{1} < d_{2} < ... < d_{\tau (n)} = n denotes the sequence of divisors of n. The function H was introduced by Erdös and has been recently investigated by *P. Erdös* and *G. Tenenbaum* [J. Number Theory 31, 285-311 (1989; Zbl 676.10030)]. In the present paper, the authors establish the upper bound

**max**_{n \leq x} H(n) \leq D(x)^{1-c+o(1)}, where D(x) = **max**_{n \leq x} \tau(n) and c = 5/3- (log 3)/(log 2), and conjecture that H(n) << \tau(n)^{1-\delta} holds for some absolute constant \delta > 0. They also show that for sufficiently large x

**max**_{n \leq x} \kappa (n) > (log x)^{(log3x)/(9 log4x)}, where log_{k}x denotes the k times iterated logarithm, thereby improving a result of *P. Erdös* and *R. R. Hall* [J. Aust. Math. Soc., Ser. A 25, 479-485 (1978; Zbl 393.10047)]. The proof of the second result depends on an estimate of independent interest, namely the bound

\#**{**n \leq x: P(n(n+1)) \leq y**}** >> xu^{-u^{7u}} for **max** (2,x^{(8 log3x)/(log2x)}) \leq y \leq x, where P(n) denotes the largest prime factor of n and u = log x/ log y. This last result represents a quantitative version of a result of the reviewer [Proc. Am. Math. Soc. 95, 517-523 (1985; Zbl 597.10056)].

**Reviewer: ** A.Hildebrand (Urbana)

**Classif.: ** * 11N25 Distribution of integers with specified multiplicative constraints

11N37 Asymptotic results on arithmetic functions

**Keywords: ** consecutive divisors; divisor function; largest prime factor

**Citations: ** Zbl 711.00008; Zbl 676.10030; Zbl 393.10047; Zbl 597.10056

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag