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) = sum1 \leq i < \tau (n)(di+1-di)-1,  \kappa(n) = sumd(d+1)|n1,

where \tau(n) is the divisor function and 1 = d1 < d2 < ... < 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

maxn \leq x H(n) \leq D(x)1-c+o(1),

where D(x) = maxn \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

maxn \leq x \kappa (n) > (log x)(log3x)/(9 log4x),

where logkx 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

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