 Gaussian Möbius function

The Gaussian Möbius function on nonzero Gaussian integers is zero when a square of a Gaussian prime divides the argument, and otherwise equals $(-1{)}^r$, where $r$ is the number of nonassociate prime factors. It takes value one on units and is unchanged by multiplying its argument by a unit.

To sum over divisors without counting associates four times, one may define it on nonzero principal ideals, or choose one generator from each associate class. Then ${\sum }_{\mathfrak{d}\mid \mathfrak{a}}\mu (\mathfrak{d})$ is one for the unit ideal and zero otherwise: for a product of $r$ distinct prime ideals the sum is $(1-1{)}^r$. This gives Möbius inversion in Gaussian arithmetic.