Prime zeta function
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)), 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 we define a sequence
(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)
The prime zeta function is related with the Artin's constant by
Specific values are:
|s||approximate value P(s)||OEIS|
The integral over the prime zeta function is usually anchored at infinity, because the pole at 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:
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 not necessarily distinct primes) define a sort of intermediate sums:
where is the total number of prime factors.
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 :
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.
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- Mathar, Richard J. (2010). "Table of Dirichlet L-series and prime zeta modulo functions for small moduli". arXiv:1008.2547.