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Prime zeta function

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The prime zeta function, usually abbreviated
P (z)
, is defined as
where the sum is over all primes
p
.

Specific values

(A085548)
(A085548)
(A085964)

Divergence of P(1)

The sum of reciprocals of primes

diverges, since by Fubini’s theorem

where
π (x)
is the prime counting function.

In particular

as proved by Mertens. In fact Mertens gave an explicit error term which was
O ( 
1
log x
)
; modern techniques improve this to

unconditionally, or

conditionally on the RH.

See also


External Links