Another generalization of the gcd-sum function

TL;DR

This paper explores a generalized gcd-sum function linked to multivariable Igusa zeta functions, revealing its asymptotic behavior and connections to the Piltz divisor function, along with a generalized Menon's identity.

Contribution

It introduces a new generalization of the gcd-sum function and establishes its asymptotic properties and relation to divisor functions, extending previous work by Kurokawa and Ochiai.

Findings

Asymptotic behavior closely connected to Piltz divisor function

Generalization of Menon's identity

New insights into gcd-sum function properties

Abstract

We investigate an arithmetic function representing a generalization of the gcd-sum function, considered by Kurokawa and Ochiai in 2009 in connection with the multivariable global Igusa zeta function for a finite cyclic group. We show that the asymptotic properties of this function are closely connected to the Piltz divisor function. A generalization of Menon's identity is also considered.