Reciprocal Riemann zeta in terms of Mobius

Theorem

The following formula holds when $\mathrm{Re}(z)>1$: $$\dfrac{1}{\zeta(z)} = \displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k^z},$$ where $\zeta$ denotes the Riemann zeta and $\mu$ denotes the Möbius function.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume III ... (previous) ... (next): $\S 17.1.2 (10)$