Difference between revisions of "Series for log(riemann zeta) over primes"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\log \left( \zeta(z) \right)=\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{kp^{ | + | $$\log \left( \zeta(z) \right)=\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{kp^{kz}},$$ |
where $\log$ denotes the [[logarithm]] and $\zeta$ denotes the [[Riemann zeta]]. | where $\log$ denotes the [[logarithm]] and $\zeta$ denotes the [[Riemann zeta]]. | ||
Latest revision as of 03:09, 1 July 2017
Theorem
The following formula holds: $$\log \left( \zeta(z) \right)=\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{kp^{kz}},$$ where $\log$ denotes the logarithm and $\zeta$ denotes the Riemann zeta.
Proof
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(2')$
