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The
prime zeta function, usually abbreviated
, is defined as
-
![{\displaystyle P(z)=\sum _{\stackrel {p}{p\,{\rm {prime}}}}{\frac {1}{p^{z}}},\quad \Re (z)>1,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/444334c67d63d089ceb837ea3dfe535e70e71b7b)
where the sum is over all
primes .
Specific values
-
(A085548)
-
(A085548)
-
(A085964)
Divergence of P(1)
The sum of reciprocals of primes
![{\displaystyle P(1)=\sum _{p}{\frac {1}{p}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7697c7a51b3f42e76d9fe65ec6b84951fd8ce87a)
diverges, since by Fubini’s theorem
-
![{\displaystyle \sum _{p}{\frac {1}{p}}=\int _{1}^{\infty }{\frac {\pi (x)}{x^{2}}}\,dx,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/4e0a12be6ba4bea00767870ed3d3558990b313ed)
where
is the
prime counting function.
In particular
-
![{\displaystyle \sum _{p<x}{\frac {1}{p}}=\log \log x+M+o(1)\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/4fe2a9137db99b3f72ef9414f46642b69c286677)
as proved by Mertens. In fact Mertens gave an explicit error term which was
; modern techniques improve this to
-
![{\displaystyle \sum _{p<x}{\frac {1}{p}}=\log \log x+M+O\left({\frac {1}{\log ^{2}x}}\right)\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/35034142a471c853c9d7fc4f2a8bfa025caa6fe6)
unconditionally, or
-
![{\displaystyle \sum _{p<x}{\frac {1}{p}}=\log \log x+M+O\left({\frac {\log x}{\sqrt {x}}}\right)\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0a09f0bcc1cf3438326130c64dc33706d34ae520)
conditionally on the RH.
See also
External Links