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In [[mathematics]], the '''Prime zeta function''' is an analogue of the [[Riemann zeta function]], studied by {{harvtxt|Glaisher|1891}}. It is defined as the following [[infinite series]], which converges for <math>\Re(s) > 1</math>:

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

== Properties ==

The [[Euler product]] for the  Riemann zeta function ζ(''s'') implies that 
:<math>\log\zeta(s)=\sum_{n>0} \frac{P(ns)}{n}</math>
which by [[Möbius inversion]] gives
:<math>P(s)=\sum_{n>0} \mu(n)\frac{\log\zeta(ns)}{n}</math>

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

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

If we define a sequence
:<math>a_n=\prod_{p^k \mid n} \frac{1}{k}=\prod_{p^k \mid \mid n} \frac{1}{k!} </math>

then

:<math>P(s)=\log\sum_{n=1}^\infty \frac{a_n}{n^s}.</math>

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

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

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

where ''L''<sub>''n''</sub> is the ''n''th [[Lucas number]].<ref>{{MathWorld|ArtinsConstant|Artin's Constant}}</ref>

Specific values are:

{{Table
|type=class="wikitable sortabel"
|hdrs=s!!approximate value P(s)!![[OEIS]]
|row1=1{{!!}}<math>\tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{5} + \tfrac{1}{7} + \tfrac{1}{11} + \cdots \to \infty. </math>
|row2=2{{!!}}<math>0{.}45224\text{ }74200\text{ }41065\text{ }49850 \ldots </math>{{!!}}{{OEIS2C|A085548}}
|row3=3{{!!}}<math>0{.}17476\text{ }26392\text{ }99443\text{ }53642 \ldots </math>{{!!}}{{OEIS2C|A085541}}
|row4=4{{!!}}<math>0{.}07699\text{ }31397\text{ }64246\text{ }84494 \ldots </math>{{!!}}{{OEIS2C|A085964}}
|row5=5{{!!}}<math>0{.}03575\text{ }50174\text{ }83924\text{ }25713 \ldots </math>{{!!}}{{OEIS2C|A085965}}
|row6=9{{!!}}<math>0{.}00200\text{ }44675\text{ }74962\text{ }45066 \ldots </math>{{!!}} {{OEIS2C|A085969}}
}}

==Analysis==

===Integral===

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

:<math>\int_s^\infty P(t)dt = \sum_p \frac{1}{p^s\log p}</math>

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

{{Table
|type=class="wikitable sortabel"
|hdrs=s!!approximate value <math>\sum _p 1/(p^s\log p)</math>!![[OEIS]]
|row1=1{{!!}}<math>1.63661632\ldots</math>{{!!}}{{OEIS2C|A137245}}
|row2=2{{!!}}<math>0.50778218\ldots</math>{{!!}}{{OEIS2C|A221711}}
|row3=3{{!!}}<math>0.22120334\ldots</math>{{!!}}
|row4=4{{!!}}<math>0.10266547\ldots</math>{{!!}}
}}

===Derivative===

The first derivative is

:<math>P'(s) \equiv \frac{d}{ds} P(s) = - \sum_p \frac{\log p}{p^s}</math>

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

{{Table
|type=class="wikitable sortabel"
|hdrs=s!!approximate value <math>P'(s)</math>!![[OEIS]]
|row1=2{{!!}}<math>-0.493091109\ldots</math>{{!!}}{{OEIS2C|A136271}}
|row2=3{{!!}}<math>-0.150757555\ldots</math>{{!!}}
|row3=4{{!!}}<math>-0.060607633\ldots</math>{{!!}}
|row4=5{{!!}}<math>-0.026838601\ldots</math>{{!!}}
}}

==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 <math>k</math> not 
necessarily distinct primes) define a sort of intermediate sums:

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

where <math>\Omega</math> is the total number of [[prime factor]]s.

{{Table
|type=class="wikitable sortabel"
|hdrs=k!!s!!approximate value <math>P_k(s)</math>!![[OEIS]]
|row1=2{{!!}}2{{!!}}<math>0.14076043434\ldots</math>{{!!}}{{OEIS2C|A117543}}
|row2=2{{!!}}3{{!!}}<math>0.02380603347\ldots</math>{{!!}}
|row3=3{{!!}}2{{!!}}<math>0.03851619298\ldots</math>{{!!}}{{OEIS2C|A131653}}
|row4=3{{!!}}3{{!!}}<math>0.00304936208\ldots</math>{{!!}}
}}

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

:<math>\zeta(s) = 1+\sum_{k=1,2,\ldots} P_k(s)</math>

===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==
{{reflist}}
*{{Cite journal
|first1=C. W.
|last1=Merrifield
|title= The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers
|journal=Proceedings of the Royal Society
|volume=33
|pages=4&ndash;10
|year=1881
|doi=10.1098/rspl.1881.0063
|jstor = 113877
}}
*{{Cite journal | last1=Fröberg | first1=Carl-Erik | title=On the prime zeta function | mr=0236123 | year=1968 | journal=Nordisk Tidskr. Informationsbehandling (BIT) | volume=8 | issue=3|pages=187&ndash;202| doi=10.1007/BF01933420}}
*{{Cite journal|last=Glaisher|first= J. W. L. |title=On the Sums of Inverse Powers of the Prime Numbers|journal=Quart. J. Math.|volume= 25|pages= 347&ndash;362|year= 1891}}
*{{Cite arXiv
|first1=Richard J.
|last1=Mathar 
|title=Twenty digits of some integrals of the prime zeta function
|eprint=0811.4739
|year=2008
|bibcode=2008arXiv0811.4739M 
}}
*{{Cite journal
|first1=Ji
|last1=Li
|title=Prime graphs and exponential composition of species
|year=2008
|volume=115
|journal=J. Combin. Theory A
|pages=1374&mdash;1401
|doi=10.1016/j.jcta.2008.02.008
|mr=2455584
}}
*{{Cite arXiv
|first1=Richard J.
|last1=Mathar
|title=Table of Dirichlet L-series and prime zeta modulo functions for small moduli
|eprint=1008.2547
|year=2010
|bibcode=2010arXiv1008.2547M 
}}

== External links ==
* [http://mathworld.wolfram.com/PrimeZetaFunction.html Prime Zeta Function, in Wolfram Mathworld]

[[Category:Zeta and L-functions]]