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
Full paper (PDF, 206 Kb)

Details

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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