= Gaussian Möbius function
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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.
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