In mathematics, the Prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series, which converges for :
.
The Euler product for the Riemann zeta function ζ(s) implies that
which by Möbius inversion gives
When s goes to 1, we have . This is used in the definition of Dirichlet density.
This gives the continuation of P(s) to , with an infinite number of logarithmic singularities at points where ns is a pole or zero of ζ(s). The line is a natural boundary as the singularities cluster near all points of this line.
If we define a sequence
then
(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)