Logarithmic derivative of Riemann zeta in terms of Mangoldt function
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Theorem
The following formula holds: $$\dfrac{\zeta'(z)}{\zeta(z)}=-\displaystyle\sum_{k=1}^{\infty} \dfrac{\Lambda(k)}{k^z},$$ where $\zeta$ denotes the Riemann zeta and $\Lambda$ denotes the Mangoldt function.
Proof
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction (2)
