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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
(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)
The prime zeta function is related to 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 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:
The first derivative is
The interesting values are again those where the sums converge slowly:
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.
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:
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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