Difference between revisions of "Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta"
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(Created page with "<div class="toccolours mw-collapsible mw-collapsed"> <strong>Theorem:</strong> The following formula holds: $$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zet...") |
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<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
$$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),$$ | ||
| − | where $\mu$ denotes the [[Möbius]] function, $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]]. | + | where $P$ denotes the [[Prime zeta function]], $\mu$ denotes the [[Möbius]] function, $\log$ denotes the [[logarithm]], and $\zeta$ denotes the [[Riemann zeta function]]. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 23:06, 6 May 2015
Theorem: The following formula holds: $$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.
Proof: █
