Prime zeta function

In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for : 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. 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 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.