Prime zeta P
From specialfunctionswiki
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.
Graph of $P(x)$ for $x>1$.
Properties
Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta
References
- Charles Watkins Merrifield: The Sums of the Series of the Reciprocals of the Prime Numbers and of Their Powers (1881)
- James Whitbread Lee Glaisher: On the Sums of the Inverse Powers of the Prime Numbers (1891)
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?
