Prime zeta function

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 \Re(s) > 1:

P(s)=\sum_{p\,\in\mathrm{\,primes}} \frac{1}{p^s}.

The Euler product for the Riemann zeta function ζ(s) implies that

\log\zeta(s)=\sum_{n>0} \frac{P(ns)}{n}

which by Möbius inversion gives

P(s)=\sum_{n>0} \mu(n)\frac{\log\zeta(ns)}{n}

When s goes to 1, we have P(s)\sim \log\zeta(s)\sim\log\left(\frac{1}{s-1}\right). This is used in the definition of Dirichlet density.

This gives the continuation of P(s) to \Re(s) > 0, with an infinite number of logarithmic singularities at points where ns is a pole or zero of ζ(s). The line \Re(s) = 0 is a natural boundary as the singularities cluster near all points of this line.

If we define a sequence

a_n=\prod_{p^k \mid n} \frac{1}{k}=\prod_{p^k \mid \mid n} \frac{1}{k!}

then

P(s)=\log\sum_{n=1}^\infty \frac{a_n}{n^s}.

(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)

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