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Prime Zeta Function

The prime zeta function

P(n)\equiv \sum_p {1\over p^n},
\end{displaymath} (1)

where the sum is taken over Primes is a generalization of the Riemann Zeta Function
\zeta(n)\equiv \sum_{k=1}^\infty {1\over k^n},
\end{displaymath} (2)

where the sum is over all integers. The prime zeta function can be expressed in terms of the Riemann Zeta Function by
$\displaystyle \ln\zeta(n)$ $\textstyle =$ $\displaystyle -\sum_{p\geq 2}\ln(1-p^{-n}) = \sum_{p\geq 2} \sum_{k=1}^\infty {p^{-kn}\over k}$  
  $\textstyle =$ $\displaystyle \sum_{k=1}^\infty {1\over k} \sum_{p\geq 2} p^{-kn}=\sum_{k=1}^\infty {P(kn)\over k}.$ (3)

Inverting then gives
P(n)=\sum_{k=1}^\infty {\mu(k)\over k}\ln\zeta(kn),
\end{displaymath} (4)

where $\mu(k)$ is the Möbius Function. The values for the first few integers starting with two are
$\displaystyle P(2)$ $\textstyle \approx$ $\displaystyle 0.452247$ (5)
$\displaystyle P(3)$ $\textstyle \approx$ $\displaystyle 0.174763$ (6)
$\displaystyle P(4)$ $\textstyle \approx$ $\displaystyle 0.0769931$ (7)
$\displaystyle P(5)$ $\textstyle \approx$ $\displaystyle 0.035755.$ (8)

See also Möbius Function, Riemann Zeta Function, Zeta Function


Hardy, G. H. and Weight, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 355-356, 1979.

© 1996-9 Eric W. Weisstein