Prime zeta function

In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by. It is defined as the following infinite series, which converges for $$\Re(s) > 1$$:


 * $$P(s)=\sum_{p\,\in\mathrm{\,primes}} \frac{1}{p^s}=\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{11^s}+\cdots.$$

Properties
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 s where ns is a pole (only ns = 1 when n is a squarefree number greater than or equal to 1), or zero of the  Riemann zeta function ζ(.). The line $$\Re(s) = 0$$ is a natural boundary as the singularities cluster near all points of this line.

If one defines 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.)

The prime zeta function is related to Artin's constant by


 * $$\ln C_{\mathrm{Artin}} = - \sum_{n=2}^{\infty} \frac{(L_n-1)P(n)}{n}$$

where Ln is the nth Lucas number.

Specific values are:

Integral
The integral over the prime zeta function is usually anchored at infinity, because the pole at $$s=1$$ prohibits defining a nice lower bound at some finite integer without entering a discussion on branch cuts in the complex plane:


 * $$\int_s^\infty P(t) \, dt = \sum_p \frac 1 {p^s\log p}$$

The noteworthy values are again those where the sums converge slowly:

Derivative
The first derivative is


 * $$P'(s) \equiv \frac{d}{ds} P(s) = - \sum_p \frac{\log p}{p^s}$$

The interesting values are again those where the sums converge slowly:

Almost-prime zeta functions
As the Riemann zeta function is a sum of inverse powers over the integers and the prime zeta function a sum of inverse powers of the prime numbers, the k-primes (the integers which are a product of $$k$$ not necessarily distinct primes) define a sort of intermediate sums:


 * $$P_k(s)\equiv \sum_{n: \Omega(n)=k} \frac 1 {n^s}$$

where $$\Omega$$ is the total number of prime factors.

Each integer in the denominator of the Riemann zeta function $$\zeta$$ may be classified by its value of the index $$k$$, which decomposes the Riemann zeta function into an infinite sum of the $$P_k$$:


 * $$\zeta(s) = 1+\sum_{k=1,2,\ldots} P_k(s)$$

Since we know that the Dirichlet series (in some formal parameter u) satisfies


 * $$P_{\Omega}(u, s) := \sum_{n \geq 1} \frac{u^{\Omega(n)}}{n^s} = \prod_{p \in \mathbb{P}} \left(1-up^{-s}\right)^{-1},$$

we can use formulas for the symmetric polynomial variants with a generating function of the right-hand-side type. Namely, we have the coefficient-wise identity that $$P_k(s) = [u^k] P_{\Omega}(u, s) = h(x_1, x_2, x_3, \ldots)$$ when the sequences correspond to $$x_j := j^{-s} \chi_{\mathbb{P}}(j)$$ where $$\chi_{\mathbb{P}}$$ denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by


 * $$P_n(s) = \sum_{{k_1+2k_2+\cdots+nk_n=n} \atop {k_1,\ldots,k_n \geq 0}} \left[\prod_{i=1}^n \frac{P(is)^{k_i}}{k_i! \cdot i^{k_i}}\right] = -[z^n]\log\left(1 - \sum_{j \geq 1} \frac{P(js) z^j}{j}\right).$$

Special cases include the following explicit expansions:


 * $$\begin{align}P_1(s) & = P(s) \\ P_2(s) & = \frac{1}{2}\left(P(s)^2+P(2s)\right) \\ P_3(s) & = \frac{1}{6}\left(P(s)^3+3P(s)P(2s)+2P(3s)\right) \\ P_4(s) & = \frac{1}{24}\left(P(s)^4+6P(s)^2 P(2s)+3 P(2s)^2+8P(s)P(3s)+6P(4s)\right).\end{align}$$

Prime modulo zeta functions
Constructing the sum not over all primes but only over primes which are in the same modulo class introduces further types of infinite series that are a reduction of the Dirichlet L-function.