Reciprocal of Riemann Zeta Function

Theorem
For $\map \Re z > 1$:


 * $\ds \frac 1 {\map \zeta z} = \sum_{k \mathop = 1}^\infty \frac{\mu \left({k}\right)} {k^z}$

where:
 * $\zeta$ is the Riemann zeta function
 * $\mu$ is the Möbius function.

Proof
By Sum of Reciprocals of Powers as Euler Product:

The expansion of this product will be:


 * $\ds 1 + \sum_{\text {$n$ prime} } \paren {\frac{-1} {n^z} } + \sum_{n \mathop = p_1 p_2} \paren {\frac {-1} {p_1^z} \frac {-1} {p_2^z} } + \sum_{n \mathop = p_1 p_2 p_3} \paren {\frac {-1} {p_1^z} \frac {-1} {p_2^z} \frac {-1}{p_3^z} } + \cdots$

which is precisely:


 * $\ds \sum_{n \mathop = 1}^\infty \frac {\map \mu n} {n^z}$

as desired.