Prime zeta function

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The prime zeta function, usually abbreviated

P (z)

, is defined as

$P(z)=\sum _{\stackrel {p}{p\,{\rm {prime}}}}{\frac {1}{p^{z}}},\quad \Re (z)>1,\,$

where the sum is over all primes

p

.

Specific values

$P(2)=\sum _{p}{\frac {1}{p^{2}}}=0.4522\ldots$ (A085548)

$P(3)=\sum _{p}{\frac {1}{p^{3}}}=0.1747\ldots$ (A085548)

$P(4)=\sum _{p}{\frac {1}{p^{4}}}=0.0769\ldots$ (A085964)

Divergence of P(1)

The sum of reciprocals of primes

$P(1)=\sum _{p}{\frac {1}{p}}$

diverges, since by Fubini’s theorem

$\sum _{p}{\frac {1}{p}}=\int _{1}^{\infty }{\frac {\pi (x)}{x^{2}}}\,dx,\,$

where

π (x)

is the prime counting function.

In particular

$\sum _{p<x}{\frac {1}{p}}=\log \log x+M+o(1)\,$

as proved by Mertens. In fact Mertens gave an explicit error term which was

O (  1 log x )

; modern techniques improve this to

$\sum _{p<x}{\frac {1}{p}}=\log \log x+M+O\left({\frac {1}{\log ^{2}x}}\right)\,$

unconditionally, or

$\sum _{p<x}{\frac {1}{p}}=\log \log x+M+O\left({\frac {\log x}{\sqrt {x}}}\right)\,$

conditionally on the RH.

See also

• Euler’s zeta function
• Riemann zeta function
• Euler products
• Euler’s alternating zeta function
• Prime zeta function
• Hurwitz zeta function
• Multivariate zeta function

• Dirichlet L-functions

External Links

• Prime zeta function—Wikipedia.org.
• MathOverflow, On the series 1 / 2 + 1 / 3 + 1 / 5 + ⋯