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**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, (*) **sum**_{d|n}h(d) \leq (2k+o(1)) **sum**_{d|n; d \leq n1/k}h(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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag