Reciprocal Riemann zeta in terms of Mobius
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Theorem
The following formula holds: $$\dfrac{1}{\zeta(z)} = \displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k^z},$$ where $\zeta$ denotes the Riemann zeta and $\mu$ denotes the Möbius function.
Proof
References
- 1953: Harry Bateman: Higher Transcendental Functions Volume III ... (previous) ... (next): $\S 17.1.2 (10)$
