Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  664.10025
Autor:  Alladi, K.; Erdös, Paul; Vaaler, J.D.
Title:  Multiplicative functions and small divisors. II. (In English)
Source:  J. Number Theory 31, No.2, 183-190 (1989).
Review:  Let k be an integer \geq 2 and h a multiplicative function satisfying 0 \leq h(p) \leq 1/(k-1) for every prime p. The authors show that, for any squarefree integer h,

(*)  sumd|nh(d) \leq (2k+o(1)) sumd|n; d \leq n1/kh(d),

where o(1) is a quantity that tends to zero as the number of prime divisors of n tends to infinity. In part I of the present paper [Analytic number theory and diophantine problems, Prog. Math. 70, 1-13 (1987; Zbl 626.10004)] the authors had obtained a similar result but under the stronger hypothesis that 0 \leq h(p) \leq c for some fixed constant c < 1/(k-1).
The proof of (*) rests on a deep theorem of Baranyai on hypergraphs. The authors give heuristic arguments suggesting that (*) remains true with the constant 4+o(1) in place of 2k+o(1) and for any real k \geq 2.
{Note: A result similar to the authors' had been obtained very recently by B. Landreau [C. R. Acad. Sci., Paris, Sér. I 307, No.14, 743- 748 (1988; Zbl 658.10053)].}
Reviewer:  A.Hildebrand
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11K65 Arithmetic functions (probabilistic number theory)
                   11A25 Arithmetic functions, etc.
Keywords:  divisors; multiplicative function; theorem of Baranyai on hypergraphs
Citations:  Zbl 626.10004; Zbl 658.10053

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