Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 626.10004
Autor: Alladi, K.; Erdös, Paul; Vaaler, J.D.
Title: Multiplicative functions and small divisors. (In English)
Source: Analytic number theory and diophantine problems, Proc. Conf., Stillwater/Okla. 1984, Prog. Math. 70, 1-13 (1987).
Review: [For the entire collection see Zbl 618.00005.]
The principal result of this paper states that if k \geq 2 and h is a nonnegative submultiplicative function satisfying 0 \leq h(p) \leq c < 1/(k-1) for all primes p, then sumd|nh(d) \leq (1-\frac{kc}{1+c})-1sumd|n, d \leq n1/kh(d) holds for all squarefree n. By writing g = 1*h, this result can be used to bound sums of the type sumn \leq x, n in Sg(n) for certain classes of multiplicative functions g and sets of integers S. The authors sketch such an application with g = euf, where f is a nonnegative additive function and u a real parameter, which leads to bounds for moments of additive functions on certain sets such as the set of shifted primes {p+1}.
Reviewer: A.Hildebrand
Classif.: * 11A25 Arithmetic functions, etc.
11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions (probabilistic number theory)
Keywords: estimates of sums of multiplicative functions; small divisors; bounds for moments of additive functions; shifted primes
Citations: Zbl 618.00005
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