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}},$$ |
where $\zeta$ denotes the [[Riemann zeta]] and $\log$ denotes the [[logarithm]]. | where $\zeta$ denotes the [[Riemann zeta]] and $\log$ denotes the [[logarithm]]. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Series for log(Riemann zeta) in terms of Mangoldt function|next=}}: § Introduction (2'') | + | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Series for log(Riemann zeta) in terms of Mangoldt function|next=Logarithmic derivative of Riemann zeta in terms of Mangoldt function}}: § Introduction $(2{'}{'})$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 02:17, 1 July 2017
Theorem
The following formula holds: $$\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 $(2{'}{'})$
