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:
$$\log \left( \zeta(z) \right)=\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{kp^{mz}},$$
+
$$\log \left( \zeta(z) \right)=\displaystyle\sum_{p \hspace{2pt} \mathrm{prime}} \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{kp^{kz}},$$
 
where $\log$ denotes the [[logarithm]] and $\zeta$ denotes the [[Riemann zeta]].
 
where $\log$ denotes the [[logarithm]] and $\zeta$ denotes the [[Riemann zeta]].
  
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==References==
 
==References==
* {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Euler product for Riemann zeta|next=Series for log(Riemann zeta) in terms of Mangoldt function}}: § Introduction (2')
+
* {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Euler product for Riemann zeta|next=Series for log(Riemann zeta) in terms of Mangoldt function}}: § Introduction $(2')$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

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