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 :

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 . This is used in the definition of Dirichlet density.

This gives the continuation of P(s) to , 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 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

where Ln is the nth Lucas number.[1]

Specific values are:

sapproximate value P(s)OEIS
1[2]
2OEIS: A085548
3OEIS: A085541
4OEIS: A085964
5OEIS: A085965
9OEIS: A085969

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 value OEIS
1OEIS: A137245
2OEIS: A221711
3
4

Derivative

The first derivative is

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

sapproximate value OEIS
2OEIS: A136271
3OEIS: A303493
4OEIS: A303494
5OEIS: A303495

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

where is the total number of prime factors.

ksapproximate value OEIS
22OEIS: A117543
23
32OEIS: A131653
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 (in some formal parameter u) 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.

See also

References

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