On an analogue of Buchstab’s identity

Debika Banerjee and Makoto Minamide
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 1, Pages 8—17
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Authors and affiliations

Debika Banerjee
Harish-Chandra Research Institute
Chhatnag Road, Jhunsi, Allahabad 211019, India

Makoto Minamide
Faculty of Science, Yamaguchi University
Yoshida 1677-1, Yamaguchi 753–8512, Japan

Abstract

In this paper, let p denote a prime. We shall consider sums of the type Φ (x,y;f)= Σ_{n≤p|n ⇒ p > y} f(n) and ψ (x,y;f)=Σ_{n≤p|n ⇒ p < y} f(n) for certain kinds of arithmetical functions f and prove some identities for Φ and ψ which are analogous to the ‘so-called’ Buchstab identity. As an application, we will prove some formulas for square-free integers.

Keywords

• Buchstab’s identity
• Square-free integers

AMS Classification

• 11N25
• 11N37

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