M\"obius functions of higher rank and Dirichlet series

TL;DR

This paper introduces higher-rank Möbius functions, extending the classical case, and evaluates related Dirichlet series using advanced number theory tools like the Riemann zeta function and cyclotomic polynomials.

Contribution

It defines a new class of higher-rank Möbius functions and computes associated Dirichlet series, expanding the understanding of arithmetic functions and their analytic properties.

Findings

Defined Möbius functions of higher rank beyond the classical case

Evaluated Dirichlet series involving sums over r-free numbers

Utilized Riemann zeta function and cyclotomic polynomials in proofs

Abstract

We introduce M\"obius functions of higher rank, a new class of arithmetic functions so that the classical M\"obius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.