NumberDB$^\beta$ Advanced search Tags Tables Help Login
Named rational Euler products over primes
edit · history · discussion · files · short url · probability theory number theory
Numbers
$P$ 
value
Artin's constant, $\prod_p(1-\frac{1}{p(p-1)})$:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: Artin's constant is the density factor in Artin's primitive-root conjecture [14] [6].
Stephens' constant, $\prod_p(1-\frac{p}{p^3-1})$:
0.5759599688929454396431633754924966925065139671764923636006407986653725516988685284364098720917261896
comment: Stephens' constant is the density factor for primes dividing terms $a^k-b$ in Stephens' two-variable Artin problem [15] [7].
rank-$2$ Artin constant, $\prod_p(1-\frac{1}{p^2(p-1)})$:
0.6975013584963659032846703508209229240731539462145153953543787528864591105960955666661548385130718796
comment: The rank-$2$ Artin constant [16] is also written $\prod_p(1-1/(p^3-p^2))$.
Feller-Tornier product part, $\prod_p(1-\frac{2}{p^2})$:
0.3226340989392446705795316925482370665709505796658327099618112524532500634862446098845234815685637552
comment: This is the product part of the Feller-Tornier constant [18].
Feller-Tornier constant, $\frac12+\frac12\prod_p(1-\frac{2}{p^2})$:
0.6613170494696223352897658462741185332854752898329163549809056262266250317431223049422617407842818776
comment: The Feller-Tornier constant is $\frac12(1+\prod_p(1-2/p^2))$, the affine transform of the product part [17] [8].
Heath-Brown-Moroz constant, $\prod_p(1-\frac1p)^7(1+\frac{7p+1}{p^2})$:
0.001317641154853178109817352322513585951250734323295251678792547421786023444096108950908340644695820646
comment: The Heath-Brown-Moroz constant is from the density of rational points on $X_0^3=X_1X_2X_3$ [19] [9].
Taniguchi's constant, $\prod_p(1-\frac{3}{p^3}+\frac{2}{p^4}+\frac{1}{p^5}-\frac{1}{p^6})$:
0.6782344919173919780355382794828948140963322391894401030364604159649833707401232332137621229334846163
comment: Taniguchi's constant is from a mean value theorem for class numbers and regulators of quadratic extensions [20] [11].
Barban's constant, $\prod_p(1+\frac{3p^2-1}{p(p+1)(p^2-1)})$:
2.596536290450542073632740656669516142373946305234501462361536498106754824578760935219371222870286431
comment: Barban's constant [21] has local factors $29/18,61/48,397/360,\ldots$ [10].
Sarnak's constant, $\prod_{p\geq3}(1-\frac{p+2}{p^3})$:
0.7236484022982000094088491498091275990417837515730770291761198897912588352514315262692852149773132909
comment: Sarnak's constant is the product over the odd primes $p\geq3$ [22] [12].
squarefree density, $\prod_p(1-\frac{1}{p^2})$:
0.6079271018540266286632767792583658334261526480334792930736541913650387257734126471472556435537310257
comment: This is the squarefree density $\prod_p(1-p^{-2})=1/\zeta(2)=6/\pi^2$ [23].
Ramanujan's product, $\prod_p(1+\frac{1}{p^2})$:
1.519817754635066571658191948145914583565381620083698232684135478412596814433531617868139108884327564
comment: $\prod_p(1+p^{-2})=15/\pi^2$, an identity of Ramanujan [24].
Landau's totient constant, $\prod_p(1+\frac{1}{p(p-1)})$:
1.943596436820759205057070362574763437187858501767805716026635688900534950693554053948179100821111301
comment: Landau's totient constant is $\prod_p(1+1/(p(p-1)))=\zeta(2)\zeta(3)/\zeta(6)$ [25].
Definition
The entries are constants given by rational Euler products $\prod_{p\in S}F(p)$, where $F(x)\in\mathbb Q(x)$ and $F(p)=1+O(p^{-2})$ as $p\to\infty$ [5].
Parameters
$P$
—   product ($P$ is one of the products listed with the entries; the Feller-Tornier constant is $\frac12+\frac12$ times one of those products)
Formulas
(1)
The Euler product for the Riemann zeta function gives $\prod_p(1-p^{-2})=1/\zeta(2)=6/\pi^2$ and $\prod_p(1+p^{-2})=\zeta(2)/\zeta(4)=15/\pi^2$ [5]; also $\prod_p(1+\frac{1}{p(p-1)})=\zeta(2)\zeta(3)/\zeta(6)$ [4].
(2)
For a rational function $F$ with $F(p)=1+O(p^{-2})$, $\log\prod_pF(p)=\sum_{j\geq2}c_jP(j)$ when $\log F(x)=\sum_{j\geq2}c_jx^{-j}$ and $P(j)$ is the prime zeta function [2].
Comments
(3)
All entries are themselves products except the Feller-Tornier constant, which is $\frac12+\frac12\prod_p(1-2/p^2)$. For all entries except Sarnak's constant, $S$ is the set of all primes; for Sarnak's constant, $S$ is the primes $p\geq3$. The condition $F(p)=1+O(p^{-2})$ gives absolute convergence because $\sum_p p^{-2}$ converges [3] [1].
(4)
The squarefree density, $\prod_p(1+p^{-2})$, and Landau's totient constant have the closed forms in (1).
Programs
(P1)
PARI/GP
default(realprecision, 140)
prodeulerrat(1 - 1/(p*(p - 1)))                       \\ Artin
prodeulerrat(1 - p/(p^3 - 1))                          \\ Stephens
(1 + prodeulerrat(1 - 2/p^2))/2                        \\ Feller-Tornier
prodeulerrat(1 - (p + 2)/p^3, 1, 3)                    \\ Sarnak
References
[1]
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003. (doi)
[2]
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, 2009. (arXiv)
Links
Similar tables
Twin prime constant —   another named rational Euler product over primes
Hafner-Sarnak-McCurley constant —   another named rational Euler product over primes
Values of the prime zeta function at rational numbers —   the logarithms of these rational Euler products can be expanded through prime zeta values
Values of the Riemann zeta function at rational numbers —   three rows have closed forms in Riemann zeta values
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds Artin's constant, Stephens' constant, the rank-$2$ Artin constant, the Feller-Tornier product part, the Feller-Tornier constant, the Heath-Brown-Moroz constant, Taniguchi's constant, Barban's constant, Sarnak's constant, the squarefree density, $\prod_p(1+p^{-2})=15/\pi^2$, and Landau's totient constant)
How they were obtained:

Values were computed with PARI's prodeulerrat [13] at $140$ decimal digits, and $100$ digits are written. The values were recomputed at $160$ decimal digits, and the $100$ written digits agree. Each row was compared with the corresponding OEIS decimal expansion named in the links.

more

The squarefree density, $\prod_p(1+p^{-2})$, and Landau's totient constant were also compared with the closed forms in (1).



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.