Logarithmic derivative of Riemann zeta in terms of series over primes

Theorem

The following formula holds: $$\dfrac{\zeta'(z)}{\zeta(z)}=-\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}}\displaystyle\sum_{k=1}^{\infty} \dfrac{\log p}{p^{mz}},$$ where $\zeta$ denotes the Riemann zeta and $\log$ denotes the logarithm.

References

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