Prime zeta function

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The prime zeta function, usually abbreviated

P (z)

, is defined as

$P(z)=\sum _{\stackrel{p}{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}}}\frac{1}{p^z},\mathrm{\Re }(z)>1,$

where the sum is over all primes

p

.

Specific values

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

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

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

Divergence of P(1)

The sum of reciprocals of primes

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

diverges, since by Fubini’s theorem

$\sum _p\frac{1}{p}=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\pi (x)}{x^2}dx,$

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 + ⋯