Prime zeta function

Template:Short description 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 $\mathrm{\Re }(s)>1$:

$P(s)=\sum _{p\in \mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}\frac{1}{p^s}=\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{11^s}+\cdots .$

Properties

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

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

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 $\mathrm{\Re }(s)>0$, 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 $\mathrm{\Re }(s)=0$ is a natural boundary as the singularities cluster near all points of this line.

If one defines a sequence

$a_n=\prod _{p^k\mid n}\frac{1}{k}=\prod _{p^k\mid \mid n}\frac{1}{k!}$

then

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

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

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

$\ln C_{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}=-\sum _{n=2}^{\mathrm{\infty }}\frac{(L_n-1)P(n)}{n}$

where L_n is the nth Lucas number.

Specific values are:

s	approximate value P(s)	OEIS
1	$\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\cdots \to \mathrm{\infty }.$
2	$0.45224742004106549850\dots$	OEIS: A085548
3	$0.17476263929944353642\dots$	OEIS: A085541
4	$0.07699313976424684494\dots$	OEIS: A085964
5	$0.03575501748392425713\dots$	OEIS: A085965
6	$0.01707008685063651295\dots$	OEIS: A085966
7	$0.00828383285613359253\dots$	OEIS: A085967
8	$0.00406140536651783056\dots$	OEIS: A085968
9	$0.00200446757496245066\dots$	OEIS: A085969

Analysis

Integral

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

$\underset{s}{\overset{\mathrm{\infty }}{\int }}P(t)dt=\sum _p\frac{1}{p^s\log p}$

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

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

Derivative

The first derivative is

$P^{\prime }(s)\equiv \frac{d}{ds}P(s)=-\sum _p\frac{\log p}{p^s}$

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

s	approximate value $P^{\prime }(s)$	OEIS
2	$-0.493091109\dots$	OEIS: A136271
3	$-0.150757555\dots$	OEIS: A303493
4	$-0.060607633\dots$	OEIS: A303494
5	$-0.026838601\dots$	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 $k$ not necessarily distinct primes) define a sort of intermediate sums:

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

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

$k$	$s$	approximate value $P_k(s)$	OEIS
2	2	$0.14076043434\dots$	OEIS: A117543
2	3	$0.02380603347\dots$
3	2	$0.03851619298\dots$	OEIS: A131653
3	3	$0.00304936208\dots$

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 $P_k$:

$\zeta (s)=1+\sum _{k=1,2,\dots }{P}_k(s)$

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

$P_{\mathrm{\Omega }}(u,s):=\sum _{n\ge 1}\frac{u^{\mathrm{\Omega }(n)}}{n^s}=\prod _{p\in \mathbb{\mathbb{P}}}{(1-u{p}^{-s})}^{-1},$

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 $P_k(s)=[u^k]P_{\mathrm{\Omega }}(u,s)=h(x_1,x_2,x_3,\dots )$ when the sequences correspond to $x_j:=j^{-s}{\chi }_{\mathbb{\mathbb{P}}}(j)$ where ${\chi }_{\mathbb{\mathbb{P}}}$ denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by

$P_n(s)=\sum _{\genfrac{}{}{0ex}{}{k_1+2k_2+\cdots +n{k}_n=n}{k_1,\dots ,k_n\ge 0}}\left[\prod _{i=1}^n\frac{P(is{)}^{k_i}}{k_i!\cdot i^{k_i}}\right]=-[z^n]\log (1-\sum _{j\ge 1}\frac{P(js)z^j}{j}).$

Special cases include the following explicit expansions:

$\begin{array}{rl}{P}_1(s)& =P(s)\\ P_2(s)& =\frac{1}{2}(P(s{)}^2+P(2s))\\ P_3(s)& =\frac{1}{6}(P(s{)}^3+3P(s)P(2s)+2P(3s))\\ P_4(s)& =\frac{1}{24}(P(s{)}^4+6P(s{)}^{2}P(2s)+3P(2s{)}^2+8P(s)P(3s)+6P(4s)).\end{array}$

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.

See also

• Divergence of the sum of the reciprocals of the primes