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 implies that

which by Möbius inversion gives

goes to 1, we have . This is used in the definition of Dirichlet density.

to is a squarefree number greater than or equal to 1), or zero of the Riemann zeta function ζ(. 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 to Artin's constant by

is the th Lucas number.

Specific values are:

Analysis

Integral

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:

Derivative

The first derivative is

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

Generalizations

Almost-prime zeta functions

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

is the total number of prime factors. 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 (in some formal parameter ) satisfies

we can use formulas for the symmetric polynomial variants with a generating function 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 that are in the same modulo class introduces further types of infinite series that are a reduction of the Dirichlet L-function.

See also

• Divergence of the sum of the reciprocals of the primes

References

External links