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 ${\displaystyle \Re (s)>1}$:

${\displaystyle P(s)=\sum _{p\,\in \mathrm {\,primes} }{\frac {1}{p^{s))}={\frac {1}{2^{s))}+{\frac {1}{3^{s))}+{\frac {1}{5^{s))}+{\frac {1}{7^{s))}+{\frac {1}{11^{s))}+\cdots .}$

Properties

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

${\displaystyle \log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n))}$

which by Möbius inversion gives

${\displaystyle P(s)=\sum _{n>0}\mu (n){\frac {\log \zeta (ns)}{n))}$

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

This gives the continuation of P(s) to ${\displaystyle \Re (s)>0}$, 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 ${\displaystyle \Re (s)=0}$ is a natural boundary as the singularities cluster near all points of this line.

If one defines a sequence

${\displaystyle a_{n}=\prod _{p^{k}\mid n}{\frac {1}{k))=\prod _{p^{k}\mid \mid n}{\frac {1}{k!))}$

then

${\displaystyle P(s)=\log \sum _{n=1}^{\infty }{\frac {a_{n)){n^{s))}.}$

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

The prime zeta function is related to Artin's constant by

${\displaystyle \ln C_{\mathrm {Artin} }=-\sum _{n=2}^{\infty }{\frac {(L_{n}-1)P(n)}{n))}$

where L_n is the nth Lucas number.^[1]

Specific values are:

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

Analysis

Integral

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

${\displaystyle \int _{s}^{\infty }P(t)\,dt=\sum _{p}{\frac {1}{p^{s}\log p))}$

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

s	approximate value ${\displaystyle \sum _{p}1/(p^{s}\log p)}$	OEIS
1	${\displaystyle 1.63661632\ldots }$	OEIS: A137245
2	${\displaystyle 0.50778218\ldots }$	OEIS: A221711
3	${\displaystyle 0.22120334\ldots }$
4	${\displaystyle 0.10266547\ldots }$

Derivative

The first derivative is

${\displaystyle P'(s)\equiv {\frac {d}{ds))P(s)=-\sum _{p}{\frac {\log p}{p^{s))))$

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

s	approximate value ${\displaystyle P'(s)}$	OEIS
2	${\displaystyle -0.493091109\ldots }$	OEIS: A136271
3	${\displaystyle -0.150757555\ldots }$	OEIS: A303493
4	${\displaystyle -0.060607633\ldots }$	OEIS: A303494
5	${\displaystyle -0.026838601\ldots }$	OEIS: A303495

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 are a product of ${\displaystyle k}$ not necessarily distinct primes) define a sort of intermediate sums:

${\displaystyle P_{k}(s)\equiv \sum _{n:\Omega (n)=k}{\frac {1}{n^{s))))$

where ${\displaystyle \Omega }$ is the total number of prime factors.

k	s	approximate value ${\displaystyle P_{k}(s)}$	OEIS
2	2	${\displaystyle 0.14076043434\ldots }$	OEIS: A117543
2	3	${\displaystyle 0.02380603347\ldots }$
3	2	${\displaystyle 0.03851619298\ldots }$	OEIS: A131653
3	3	${\displaystyle 0.00304936208\ldots }$

Each integer in the denominator of the Riemann zeta function ${\displaystyle \zeta }$ may be classified by its value of the index ${\displaystyle k}$, which decomposes the Riemann zeta function into an infinite sum of the ${\displaystyle P_{k))$:

${\displaystyle \zeta (s)=1+\sum _{k=1,2,\ldots }P_{k}(s)}$

Since we know that the Dirichlet series (in some formal parameter u) satisfies

${\displaystyle P_{\Omega }(u,s):=\sum _{n\geq 1}{\frac {u^{\Omega (n))){n^{s))}=\prod _{p\in \mathbb {P} }\left(1-up^{-s}\right)^{-1},}$

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 ${\displaystyle P_{k}(s)=[u^{k}]P_{\Omega }(u,s)=h(x_{1},x_{2},x_{3},\ldots )}$ when the sequences correspond to ${\displaystyle x_{j}:=j^{-s}\chi _{\mathbb {P} }(j)}$ where ${\displaystyle \chi _{\mathbb {P} ))$ denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by

${\displaystyle P_{n}(s)=\sum _((k_{1}+2k_{2}+\cdots +nk_{n}=n} \atop {k_{1},\ldots ,k_{n}\geq 0))\left[\prod _{i=1}^{n}{\frac {P(is)^{k_{i))}{k_{i}!\cdot i^{k_{i))))\right]=-[z^{n}]\log \left(1-\sum _{j\geq 1}{\frac {P(js)z^{j)){j))\right).}$

Special cases include the following explicit expansions:

${\displaystyle {\begin{aligned}P_{1}(s)&=P(s)\\P_{2}(s)&={\frac {1}{2))\left(P(s)^{2}+P(2s)\right)\\P_{3}(s)&={\frac {1}{6))\left(P(s)^{3}+3P(s)P(2s)+2P(3s)\right)\\P_{4}(s)&={\frac {1}{24))\left(P(s)^{4}+6P(s)^{2}P(2s)+3P(2s)^{2}+8P(s)P(3s)+6P(4s)\right).\end{aligned))}$

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.