Prime zeta function

Properties

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

which by Möbius inversion gives

When s goes to 1, we have P ( s ) ∼ log ⁡ ζ ( s ) ∼ log ⁡ ( 1 s − 1 ) . This is used in the definition of Dirichlet density.

This gives the continuation of P(s) to ℜ ( s ) > 0 , with an infinite number of logarithmic singularities at points s where ns is a pole (only ns=1)), or zero of the Riemann zeta function ζ(.). The line ℜ ( s ) = 0 is a natural boundary as the singularities cluster near all points of this line.

If one defines a sequence

then

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

The prime zeta function is related with the Artin's constant by

where L_n 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 to define a nice lower bound at some finite integer without entering a discussion on branch cuts in the complex plane:

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

Derivative

The first derivative is

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 a are a product of k not necessarily distinct primes) define a sort of intermediate sums:

where Ω is the total number of prime factors.

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

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.