Prime zeta P

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.

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  Graph of $P(x)$ for $x>1$.

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.

References

• Charles Watkins Merrifield: The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers (1881)

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