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Sums involving common divisors. (English) Zbl 0602.10041
The authors give a uniform upper bound for the sum $(*)\quad \sum_{1\leq r,s\leq k}(\frac{f(r) g(s) (a_ r,a_ s)^ 2}{a_ r a_ s})^{1/2},$ where $$\{a_ k\}$$ is an arbitrary sequence of distinct positive integers and f and g are arbitrary non-negative functions. Sums of this type arise in certain diophantine problems; in fact the authors establish here a bound for the number of solutions of the system $ma_ r=nb_ s,\quad 1\leq m\leq M,\quad 1\leq n\leq N,\quad 1\leq r\leq R,\quad 1\leq s\leq S,$ where $$\{a_ r\}$$ and $$\{b_ s\}$$ are arbitrary sequences of distinct positive integers that is essentially equivalent to their bound for (*). An application of these results to the metric theory of diophantine approximation is announced in the paper; this has appeared in the meantime [the second author, Math. Proc. Camb. Philos. Soc. 99, 385-394 (1986; see the following review)].
[Note: There is a misprint in the statement of Theorem $$1: (a_ r,a_ s)$$ should be replaced by $$(a_ r,a_ s)^ 2.]$$
Reviewer: A.Hildebrand

##### MSC:
 11K60 Diophantine approximation in probabilistic number theory 11B75 Other combinatorial number theory
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