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On an analogue of Buchstab’s identity. (English) Zbl 1376.11069

Summary: In this paper, let \(p\) denote a prime. We shall consider sums of the type \(\Phi(x,y;f)=\sum_{n\geq x, p\mid n\Rightarrow p>y} f(n)\) and \(\psi(x,y;f)=\sum_{n\geq x, p\mid n\Rightarrow p<y} f(n)\) for certain kinds of arithmetical functions \(f\) and prove some identities for \(\Phi\) and \(\psi\) which are analogous to the ‘so-called’ Buchstab identity. As an application, we prove some formulas for square-free integers.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions
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