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 $\Re(s) > 1$:

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

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 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.)

The prime zeta function is related with the 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.[1]

Specific values are:

s approximate value P(s) OEIS
1 $\tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{5} + \tfrac{1}{7} + \tfrac{1}{11} + \cdots \to \infty.$
2 $0{.}45224\text{ }74200\text{ }41065\text{ }49850 \ldots$
3 $0{.}17476\text{ }26392\text{ }99443\text{ }53642 \ldots$
4 $0{.}07699\text{ }31397\text{ }64246\text{ }84494 \ldots$
5 $0{.}03575\text{ }50174\text{ }83924\text{ }25713 \ldots$
9 $0{.}00200\text{ }44675\text{ }74962\text{ }45066 \ldots$

Analysis

Integral

The integral over the prime zeta function is usually anchored at infinity, because the pole at $s=1$ prohibits to define 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:

s approximate value $\sum _p 1/(p^s\log p)$ OEIS
1 $1.63661632\ldots$
2 $0.50778218\ldots$
3 $0.22120334\ldots$
4 $0.10266547\ldots$

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:

s approximate value $P'(s)$ OEIS
2 $-0.493091109\ldots$
3 $-0.150757555\ldots$
4 $-0.060607633\ldots$
5 $-0.026838601\ldots$

Generalizations

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 a 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.

k s approximate value $P_k(s)$ OEIS
2 2 $0.14076043434\ldots$
2 3 $0.02380603347\ldots$
3 2 $0.03851619298\ldots$
3 3 $0.00304936208\ldots$

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 $P_k$:

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

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.