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 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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0f349e0d4a9b2e346f3706962bd88deee837f7)
Properties
The Euler product for the Riemann zeta function ζ(s) implies that
![{\displaystyle \log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d24acd2308a46caae95dbf5baabad6bd48a68b31)
which by Möbius inversion gives
![{\displaystyle P(s)=\sum _{n>0}\mu (n){\frac {\log \zeta (ns)}{n))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35746b72820f6d4397217be901a91b5466d8d364)
When s goes to 1, we have
.
This is used in the definition of Dirichlet density.
This gives the continuation of P(s) to
, 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
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!))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0002fa70b34359d880fddf50fba0958f289897)
then
![{\displaystyle P(s)=\log \sum _{n=1}^{\infty }{\frac {a_{n)){n^{s))}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d05109a18304b5d8a9394e311f35329efd60678)
(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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5874d402516728b9696b8a6cf92d122b6051a79d)
where Ln is the nth Lucas number.[1]
Specific values are:
s |
approximate value P(s) |
OEIS
|
1 |
[2] |
|
2 |
![{\displaystyle 0{.}45224{\text{ ))74200{\text{ ))41065{\text{ ))49850\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea9213b175fefc4359d4357e5ea1d251f0306b4) |
OEIS: A085548
|
3 |
![{\displaystyle 0{.}17476{\text{ ))26392{\text{ ))99443{\text{ ))53642\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ebe7568ca62985f781034cba56d69e7aa7c3408) |
OEIS: A085541
|
4 |
![{\displaystyle 0{.}07699{\text{ ))31397{\text{ ))64246{\text{ ))84494\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a453198dd036c8277f3932b44ed7baafb6cf310) |
OEIS: A085964
|
5 |
![{\displaystyle 0{.}03575{\text{ ))50174{\text{ ))83924{\text{ ))25713\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/88c186122510f1a31b0b291b3f020a4bb7311af3) |
OEIS: A085965
|
9 |
![{\displaystyle 0{.}00200{\text{ ))44675{\text{ ))74962{\text{ ))45066\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf2c641afffe458081c0a081a052367cd864f7e6) |
OEIS: A085969
|
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
not
necessarily distinct primes) define a sort of intermediate sums:
![{\displaystyle P_{k}(s)\equiv \sum _{n:\Omega (n)=k}{\frac {1}{n^{s))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0aeb241b49adbdd58fb90255cb9b4dd388903f)
where
is the total number of prime factors.
k |
s |
approximate value ![{\displaystyle P_{k}(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b11149251d46e8952de34ff8678622a65fe04c82) |
OEIS
|
2 |
2 |
![{\displaystyle 0.14076043434\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63bec847499cda889ddf4bc74af43a3af470e467) |
OEIS: A117543
|
2 |
3 |
![{\displaystyle 0.02380603347\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b46149d8ad55d3f2fc19da09addd64ba823519) |
|
3 |
2 |
![{\displaystyle 0.03851619298\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc98a4ea16adf5f02b05bf78e740fc8e178db6d) |
OEIS: A131653
|
3 |
3 |
![{\displaystyle 0.00304936208\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/76b151508b8be0338efa8321d8c2749154df8ec0) |
|
Each integer in the denominator of the Riemann zeta function
may be classified by its value of the index
, which decomposes the Riemann zeta
function into an infinite sum of the
:
![{\displaystyle \zeta (s)=1+\sum _{k=1,2,\ldots }P_{k}(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f81c59cc50f1ba0b26d4100cbfb1fe9ebe284743)
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78dc80e5c6cfc252558c9ec9047a5dc6d3878506)
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
when the sequences correspond to
where
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ae44734b9408a6228db71ff237f10aefcd10f7)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65b708b6d7616fb8b37919298dd9d13b69fec93e)
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.