Difference between revisions of "Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta"
From specialfunctionswiki
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zeta(kz),$$ | $$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zeta(kz),$$ | ||
where $P$ denotes the [[Prime zeta function]], $\mu$ denotes the [[Möbius]] function, $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]]. | where $P$ denotes the [[Prime zeta function]], $\mu$ denotes the [[Möbius]] function, $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 19:31, 15 June 2016
Theorem
The following formula holds: $$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zeta(kz),$$ where $P$ denotes the Prime zeta function, $\mu$ denotes the Möbius function, $\log$ denotes the logarithm, and $\zeta$ denotes the Riemann zeta function.
