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

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

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
6	${\displaystyle 0{.}01707{\text{ }}00868{\text{ }}50636{\text{ }}51295\ldots }$	OEIS: A085966
7	${\displaystyle 0{.}00828{\text{ }}38328{\text{ }}56133{\text{ }}59253\ldots }$	OEIS: A085967
8	${\displaystyle 0{.}00406{\text{ }}14053{\text{ }}66517{\text{ }}83056\ldots }$	OEIS: A085968
9	${\displaystyle 0{.}00200{\text{ }}44675{\text{ }}74962{\text{ }}45066\ldots }$	OEIS: A085969

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	${\displaystyle 0.10266547\ldots }$

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

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:

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

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.

• Divergence of the sum of the reciprocals of the primes

• Merrifield, C. W. (1881). "The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers". Proceedings of the Royal Society. 33 (216–219): 4–10. doi:10.1098/rspl.1881.0063. JSTOR 113877.
• Fröberg, Carl-Erik (1968). "On the prime zeta function". Nordisk Tidskr. Informationsbehandling (BIT). 8 (3): 187–202. doi:10.1007/BF01933420. MR 0236123. S2CID 121500209.
• Glaisher, J. W. L. (1891). "On the Sums of Inverse Powers of the Prime Numbers". Quart. J. Math. 25: 347–362.
• Mathar, Richard J. (2008). "Twenty digits of some integrals of the prime zeta function". arXiv:0811.4739 [math.NT].
• Li, Ji (2008). "Prime graphs and exponential composition of species". Journal of Combinatorial Theory. Series A. 115 (8): 1374–1401. arXiv:0705.0038. doi:10.1016/j.jcta.2008.02.008. MR 2455584. S2CID 6234826.
• Mathar, Richard J. (2010). "Table of Dirichlet L-series and prime zeta modulo functions for small moduli". arXiv:1008.2547 [math.NT].

• Weisstein, Eric W. "Prime Zeta Function". MathWorld.