In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891).
: The Euler product for the Riemann zeta function ζ(s) implies that which by Möbius inversion gives When s goes to 1, we have
, 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(.).
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: 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
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
, 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.
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.