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 |
Analysis
Integral
The integral over the prime zeta function is usually anchored at infinity, because the pole at
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb30aa809c2ca7650dbc8235717795d472f33a0)
The noteworthy values are again those where the sums converge slowly:
s | approximate value ![{\displaystyle \sum _{p}1/(p^{s}\log p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0c2ec371b78a8314b832c84502dbe75909eb4) | OEIS |
1 | ![{\displaystyle 1.63661632\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2506e9565ba703099f01ada6d6f69bd384e8e2d7) | OEIS: A137245 |
2 | ![{\displaystyle 0.50778218\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a52caf4df08965c3fc900d32d8bb4aadbfa18491) | OEIS: A221711 |
3 | ![{\displaystyle 0.22120334\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8afef701a6fecd36221c91b149d431eb8b5efd50) | |
4 | ![{\displaystyle 0.10266547\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89704982a778e9bf6049046eafdf32bc97068e1e) | |
Derivative
The first derivative is
![{\displaystyle P'(s)\equiv {\frac {d}{ds}}P(s)=-\sum _{p}{\frac {\log p}{p^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08bf682fa55c0e3ea380d0d803fef0e542b423f6)
The interesting values are again those where the sums converge slowly:
s | approximate value ![{\displaystyle P'(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2971cd076b22da11ccce5daaa0ff9199357d551d) | OEIS |
2 | ![{\displaystyle -0.493091109\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1051981bee465e48a8402c2e19b80a2bcc16aca) | OEIS: A136271 |
3 | ![{\displaystyle -0.150757555\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/adde0a44f7ede924f7628abacf1ff9859778f3c3) | OEIS: A303493 |
4 | ![{\displaystyle -0.060607633\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e99b4ffc4a09695936fadf9a94609d0f5efe2c) | OEIS: A303494 |
5 | ![{\displaystyle -0.026838601\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc973c336e6cbdf602c52d3e73fdd3f96531279d) | 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
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.
See also
References
- 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].
External links