 Möbius divisor-sum identity

For $n>1$, expand ${\prod }_{p\mid n}(1-1)=0$ over the subsets of its distinct prime divisors. The coefficient of a squarefree divisor is its Möbius function value; nonsquarefree divisors have value zero. At $n=1$ the empty product is one. This identity is the multiplicative cancellation behind Möbius inversion and reciprocal Dirichlet series.