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 ⁠${\displaystyle \Re (s)>1}$⁠:

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

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

This gives the continuation of ${\displaystyle P(s)}$ to ⁠${\displaystyle \Re (s)>0}$⁠, with an infinite number of logarithmic singularities at points ${\displaystyle s}$ where ${\displaystyle ns}$ is a pole (only ${\displaystyle ns=1}$ when ${\displaystyle n}$ 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 ${\displaystyle L_{n}}$ is the ⁠${\displaystyle n}$⁠th Lucas number.^[1]

Specific values are:

${\displaystyle s}$	approximate value ${\displaystyle P(s)}$	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:

${\displaystyle s}$	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:

${\displaystyle s}$	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 ${\displaystyle k}$-primes (the integers that 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.

${\displaystyle k}$	${\displaystyle s}$	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 ⁠${\displaystyle u}$⁠) 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 ${\displaystyle P_{k}(s)=P_{\Omega }(u,s)=h(x_{1},x_{2},x_{3},\ldots )}$ 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

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

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 that 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 .
• 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 .

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