Logarithmic derivative of Riemann zeta in terms of Mangoldt function

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.

References

• 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(2{'}{'})$