Difference between revisions of "Prime zeta P"

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Fröberg, Carl-Erik . On the prime zeta function. Nordisk Tidskr. Informationsbehandling (BIT)  8  1968 187--202.<br />
 
Fröberg, Carl-Erik . On the prime zeta function. Nordisk Tidskr. Informationsbehandling (BIT)  8  1968 187--202.<br />
 
[http://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1/49434#49434 How does ∑p<xp−s grow asymptotically for Re(s)<1?] <br />
 
[http://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1/49434#49434 How does ∑p<xp−s grow asymptotically for Re(s)<1?] <br />
[http://rspl.royalsocietypublishing.org/content/33/216-219/4.full.pdf+html The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers]
+
[http://rspl.royalsocietypublishing.org/content/33/216-219/4.full.pdf+html The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers]<br />
[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0025%7CLOG_0038 On the sums of the inverse powers of the prime numbers - J.W.L. Glaisher]
+
[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0025%7CLOG_0038 On the sums of the inverse powers of the prime numbers - J.W.L. Glaisher]<br />
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Revision as of 09:15, 1 June 2016

The prime zeta function is defined by $$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$ where $\mathrm{Re}(z)>1$. It can be extended outside of this domain via analytic continuation.


Properties

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

References

References

Fröberg, Carl-Erik . On the prime zeta function. Nordisk Tidskr. Informationsbehandling (BIT) 8 1968 187--202.
How does ∑p<xp−s grow asymptotically for Re(s)<1?
The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers
On the sums of the inverse powers of the prime numbers - J.W.L. Glaisher