New inversion properties of \(\mu\) and \(\mu^*\) (Q2548268)

Let \(\mu(n)\) denote the Möbius function and \(\mu^*(n)\) denote the unitary analogue of \(\mu(n)\), which is defined by \(\mu^*(n)=(-1)^{\omega(n)}\), where \(\omega(n)\) is the number of distinct prime divisors of \(n\), \(\omega(1)=0\). The author proves four inversion formulae of the type: \[ g(n)=\sum_{_{\substack{ d^k\delta(n)\\ (d,\delta)=1}}} f(\delta) \leftrightarrow f(n)=\sum_{_{\substack{ d^k\delta(n)\\ (d,\delta)=1}}} \mu^*(d)g(\delta). \]