Prime zeta function

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 \Re(s) > 1:

P(s)=\sum_{p\,\in\mathrm{\,primes}} \frac{1}{p^s}.

Properties

The Euler product for the Riemann zeta function ζ(s) implies that

\log\zeta(s)=\sum_{n>0} \frac{P(ns)}{n}

which by Möbius inversion gives

P(s)=\sum_{n>0} \mu(n)\frac{\log\zeta(ns)}{n}

When s goes to 1, we have P(s)\sim \log\zeta(s)\sim\log\left(\frac{1}{s-1}\right). This is used in the definition of Dirichlet density.

This gives the continuation of P(s) to \Re(s) > 0, with an infinite number of logarithmic singularities at points where ns is a pole or zero of ζ(s). The line \Re(s) = 0 is a natural boundary as the singularities cluster near all points of this line.

If we define a sequence

a_n=\prod_{p^k \mid n} \frac{1}{k}=\prod_{p^k \mid \mid n} \frac{1}{k!}

then

P(s)=\log\sum_{n=1}^\infty \frac{a_n}{n^s}.

(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)

The prime zeta function is related with the Artin's constant by

\ln C_{\mathrm{Artin}} = - \sum_{n=2}^{\infty} \frac{(L_n-1)P(n)}{n}

where Ln is the nth Lucas number.[1]

Specific values are:

sapproximate value P(s)OEIS
1 \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{5} + \tfrac{1}{7} + \tfrac{1}{11} + \cdots \to \infty.
2 0{.}45224\text{ }74200\text{ }41065\text{ }49850 \ldots  A085548
3 0{.}17476\text{ }26392\text{ }99443\text{ }53642 \ldots  A085541
4 0{.}07699\text{ }31397\text{ }64246\text{ }84494 \ldots  A085964
5 0{.}03575\text{ }50174\text{ }83924\text{ }25713 \ldots  A085965
9 0{.}00200\text{ }44675\text{ }74962\text{ }45066 \ldots  A085969

Analysis

Integral

The integral over the prime zeta function is usually anchored at infinity, because the pole at s=1 prohibits to define a nice lower bound at some finite integer without entering a discussion on branch cuts in the complex plane:

\int_s^\infty P(t)dt = \sum_p \frac{1}{p^s\log p}

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

sapproximate value \sum _p 1/(p^s\log p)OEIS
1 1.63661632\ldots  A137245
2 0.50778218\ldots  A221711
3 0.22120334\ldots
4 0.10266547\ldots

Derivative

The first derivative is

P'(s) \equiv \frac{d}{ds} P(s) = - \sum_p \frac{\log p}{p^s}

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

sapproximate value P'(s)OEIS
2 -0.493091109\ldots  A136271
3 -0.150757555\ldots
4 -0.060607633\ldots
5 -0.026838601\ldots

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 a are a product of k not necessarily distinct primes) define a sort of intermediate sums:

P_k(s)\equiv \sum_{n: \Omega(n)=k} \frac{1}{n^s}

where \Omega is the total number of prime factors.

ksapproximate value P_k(s)OEIS
2 2 0.14076043434\ldots  A117543
2 3 0.02380603347\ldots
3 2 0.03851619298\ldots  A131653
3 3 0.00304936208\ldots

Each integer in the denominator of the Riemann Zeta Function \zeta may be classified by its value of the index k, which decomposes the Riemann Zeta Function into an infinite sum of the P_k:

\zeta(s) = 1+\sum_{k=1,2,\ldots} P_k(s)

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.

References

  1. ↑ Weisstein, Eric W., "Artin's Constant", MathWorld.

External links