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Prime zeta function

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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:

s approximate value P(s) OEIS
1 [2]
2 OEISA085548
3 OEISA085541
4 OEISA085964
5 OEISA085965
9 OEISA085969

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:

s approximate value OEIS
1 OEISA137245
2 OEISA221711
3
4

Derivative

The first derivative is

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

s approximate value OEIS
2 OEISA136271
3 OEISA303493
4 OEISA303494
5 OEISA303495

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.

k s approximate value OEIS
2 2 OEISA117543
2 3
3 2 OEISA131653
3 3

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

  1. ^ Weisstein, Eric W. "Artin's Constant". MathWorld.
  2. ^ See divergence of the sum of the reciprocals of the primes.
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Prime zeta function
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