A generalization of Apostol's M\"obius functions of order k

TL;DR

This paper extends Apostol's Möbius functions of order k by introducing a second parameter m and derives asymptotic formulas for their partial sums, broadening the understanding of these number-theoretic functions.

Contribution

The paper introduces a generalized form of Apostol's Möbius functions with an additional parameter and establishes asymptotic formulas for their partial sums, advancing the theoretical framework.

Findings

Derived asymptotic formulas for the generalized functions

Extended the class of Apostol's Möbius functions

Provided new insights into their asymptotic behavior

Abstract

Apostol's Mobius functions of order k are generalized to depend on a second integer parameter m. Asymptotic formulas are obtained for the partial sums of these generalized functions.