A remark on the binomial theorem and Möbius inversion (Q1312716)

For \(z\in\mathbb{C}\), \(D_ z\) is the multiplicative arithmetical function satisfying \((\zeta(s))^ z= \sum^ \infty_{n=1} D_ z(n)n^{-s}\), where \(\zeta\) is the Riemann zeta-function. In particular, \(D_{-1}\) is the Möbius function \(\mu\). The author proves that \(D_{z+w}= D_ z * D_ w\), where \(*\) denotes the Dirichlet convolution. This implies that the Dirichlet inverse of \(D_ z\) is \(D_{-z}\), which gives the inversion formula \[ F= (\chi\cdot D_ z) * G\Leftrightarrow G= (\chi\cdot D_{-z}) * F \] for any completely multiplicative function \(\chi\). With \(z=1\) and \(\chi\equiv 1\), this inversion formula is the classical Möbius inversion formula. The author also notes that the equation \(D_{z+ w}= D_ z* D_ w\) at \(p^ n\) means that \[ (z+ w)^{[n]}= \sum^ n_{j=0}{n\choose j} z^{[n-j]} w^{[j]}, \] where \(z^{[n]}\) is the rising factorial.