Difference between revisions of "Prime zeta P"
(→References) |
|||
| Line 1: | Line 1: | ||
| + | __NOTOC__ | ||
The prime zeta function is defined by | The prime zeta function is defined by | ||
$$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$ | $$P(z) = \displaystyle\sum_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p^z},$$ | ||
Revision as of 17:27, 15 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.
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.
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
