Difference between revisions of "Logarithmic derivative of Riemann zeta in terms of series over primes"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\dfrac{\zeta'(z)}{\zeta(z)}=-\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}}\displaystyle\sum_{k=1}^{\infty} \dfrac{\log p}{p^{ | + | $$\dfrac{\zeta'(z)}{\zeta(z)}=-\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}}\displaystyle\sum_{k=1}^{\infty} \dfrac{\log p}{p^{kz}},$$ |
where $\zeta$ denotes the [[Riemann zeta]] and $\log$ denotes the [[logarithm]]. | where $\zeta$ denotes the [[Riemann zeta]] and $\log$ denotes the [[logarithm]]. | ||
Latest revision as of 02:17, 1 July 2017
Theorem
The following formula holds: $$\dfrac{\zeta'(z)}{\zeta(z)}=-\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}}\displaystyle\sum_{k=1}^{\infty} \dfrac{\log p}{p^{kz}},$$ where $\zeta$ denotes the Riemann zeta and $\log$ denotes the logarithm.
Proof
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(2{'}{'})$
