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 :

Properties

The Euler product for the Riemann zeta function ζ 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 to, with an infinite number of logarithmic singularities at points s where ns is a pole, or zero of the Riemann zeta function ζ. The line is a natural boundary as the singularities cluster near all points of this line.
If one defines a sequence
then
The prime zeta function is related to Artin's constant by
where Ln is the nth Lucas number.
Specific values are:
sapproximate value POEIS
1
2
3
4
5
9

Analysis

Integral

The integral over the prime zeta function is usually anchored at infinity,
because the pole at prohibits defining 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:
sapproximate valueOEIS
1
2
3
4

Derivative

The first derivative is
The interesting values are again those where the sums converge slowly:
sapproximate valueOEIS
2
3
4
5

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 define a sort of intermediate sums:
where is the total number of prime factors.
ksapproximate valueOEIS
22
23
32
33

Each integer in the denominator of the Riemann zeta function
may be classified by its value of the index, which decomposes the Riemann zeta
function into an infinite sum of the :
Since we know that the Dirichlet series satisfies
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 when the sequences correspond to where denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by
Special cases include the following explicit expansions:

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.