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Densities of primes with a given primitive root
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Numbers
$a$ 
$\delta(a)$
2:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the primes for which $2$ is a primitive root form OEIS A001122 [7] and the quadratic-field discriminant is $d=8$.
-2:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-8$.
3:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=12$.
-3:
0.4487469763430427456656736652156996981339550983273800505136913629008202184480336988131651805047006828
comment: This is $\frac{6}{5}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-3$.
-4:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-4$.
5:
0.3936376985465287242681347940488593843280307880064737285207818972814212442526611393097940179865795463
comment: This is $\frac{20}{19}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=5$.
-5:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-20$.
6:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=24$.
-6:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-24$.
7:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=28$.
-7:
0.3830766871221096609341116654280363276753275229623976040970536024763099425775897428892873492113298512
comment: This is $\frac{42}{41}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-7$.
8:
0.2243734881715213728328368326078498490669775491636900252568456814504101092240168494065825902523503414
comment: This is $\frac{3}{5}A$, where $A$ is Artin's constant; $8=2^3$ and the quadratic-field discriminant is $d=8$.
-8:
0.2243734881715213728328368326078498490669775491636900252568456814504101092240168494065825902523503414
comment: This is $\frac{3}{5}A$, where $A$ is Artin's constant; $-8=(-2)^3$ and the quadratic-field discriminant is $d=-8$.
-9:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-4$.
10:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=40$.
-10:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-40$.
11:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=44$.
-11:
0.3773866009001124007891751007165670244245799756575826112882725528982433029761751289713162832990602072
comment: This is $\frac{110}{109}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-11$.
12:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=12$.
-12:
0.4487469763430427456656736652156996981339550983273800505136913629008202184480336988131651805047006828
comment: This is $\frac{6}{5}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-3$.
13:
0.3763684317715842383002424288905868435962204050487703649469669495297201832144798764239449901007167017
comment: This is $\frac{156}{155}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=13$.
-13:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-52$.
14:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=56$.
-14:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-56$.
15:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=60$.
-15:
0.3700194366337370008120467064059278212683489407260853048095349834445359695975014709512063769073847735
comment: This is $\frac{94}{95}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-15$.
-16:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-4$.
17:
0.3753357243705646581213506670930821583408234524755454297039484954152001827117625033123644806189378405
comment: This is $\frac{272}{271}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=17$.
-17:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-68$.
18:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=8$.
-18:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-8$.
19:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=76$.
-19:
0.3750524582339213563481436791392211553318979560800683706639355965593365462102334432895955320933715384
comment: This is $\frac{342}{341}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-19$.
20:
0.3936376985465287242681347940488593843280307880064737285207818972814212442526611393097940179865795463
comment: This is $\frac{20}{19}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=5$.
-20:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-20$.
21:
0.3721316389186208134788513321300924325988895937349005296942806424055582299325157502353077106624347125
comment: This is $\frac{204}{205}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=21$.
-21:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-84$.
22:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=88$.
-22:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-88$.
23:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=92$.
-23:
0.3746963201808244708033512782164093189039295045439840025741383327191667170539687320123128404874233424
comment: This is $\frac{506}{505}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-23$.
24:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=24$.
-24:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-24$.
-25:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-4$.
26:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=104$.
-26:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-104$.
27:
0.2243734881715213728328368326078498490669775491636900252568456814504101092240168494065825902523503414
comment: This is $\frac{3}{5}A$, where $A$ is Artin's constant; $27=3^3$ and the quadratic-field discriminant is $d=12$.
-27:
0.4487469763430427456656736652156996981339550983273800505136913629008202184480336988131651805047006828
comment: This is $\frac{6}{5}A$, where $A$ is Artin's constant; $-27=(-3)^3$ and the quadratic-field discriminant is $d=-3$.
28:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=28$.
-28:
0.3830766871221096609341116654280363276753275229623976040970536024763099425775897428892873492113298512
comment: This is $\frac{42}{41}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-7$.
29:
0.3744169181982641897662628608252652639588692353491909176090396492760645472459960577849261473179376844
comment: This is $\frac{812}{811}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=29$.
-29:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-116$.
30:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=120$.
-30:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-120$.
31:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=124$.
-31:
0.3743583494788569729719021426718700388092736288522277062950600713112332285223101362542551290539752736
comment: This is $\frac{930}{929}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-31$.
32:
0.2952282739098965432011010955366445382460230910048552963905864229610659331894958544823455134899346597
comment: This is $\frac{15}{19}A$, where $A$ is Artin's constant; $32=2^5$ and the quadratic-field discriminant is $d=8$.
-32:
0.2952282739098965432011010955366445382460230910048552963905864229610659331894958544823455134899346597
comment: This is $\frac{15}{19}A$, where $A$ is Artin's constant; $-32=(-2)^5$ and the quadratic-field discriminant is $d=-8$.
33:
0.3732696561630202655078386450723862932490391031958635282560368523211715578527986730189019238448886413
comment: This is $\frac{544}{545}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=33$.
-33:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-132$.
34:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=136$.
-34:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-136$.
35:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=140$.
-35:
0.3734757676453650579031815485000153670819609183768738546209369708352996683275248370524631048701937647
comment: This is $\frac{778}{779}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-35$.
-36:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-4$.
37:
0.3742367721568575865431237929296969683912022232482282915628831050487681761662790425864863639070005694
comment: This is $\frac{1332}{1331}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=37$.
-37:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-148$.
38:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=152$.
-38:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-152$.
39:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=156$.
-39:
0.3734732899887258980056251794375823294147110173176259775242979729948761818051377235283761824845573424
comment: This is $\frac{774}{775}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-39$.
40:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=40$.
-40:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-40$.
41:
0.3741839745793116244110762715851878711306113286846162715286017058965553987465808755611098718871817774
comment: This is $\frac{1640}{1639}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=41$.
-41:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-164$.
42:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=168$.
-42:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-168$.
43:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=172$.
-43:
0.3741629913552794084359218095011789726823282121787850282676484771001298774317400092597305244651382424
comment: This is $\frac{1806}{1805}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-43$.
44:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=44$.
-44:
0.3773866009001124007891751007165670244245799756575826112882725528982433029761751289713162832990602072
comment: This is $\frac{110}{109}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-11$.
45:
0.3936376985465287242681347940488593843280307880064737285207818972814212442526611393097940179865795463
comment: This is $\frac{20}{19}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=5$.
-45:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-20$.
46:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=184$.
-46:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-184$.
47:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=188$.
-47:
0.3741288611960737375170393583974790788853967771802389592821998791422078174782696501750976092284293059
comment: This is $\frac{2162}{2161}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-47$.
48:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=12$.
-48:
0.4487469763430427456656736652156996981339550983273800505136913629008202184480336988131651805047006828
comment: This is $\frac{6}{5}A$, where $A$ is Artin's constant; the quadratic-field discriminant is $d=-3$.
-49:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-4$.
50:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=8$.
-50:
0.3739558136192022880547280543464164151116292486061500420947428024173501820400280823443043170872505690
comment: This is Artin's constant $A$; the quadratic-field discriminant is $d=-8$.
Definition
For $a\in\mathbb Z$ with $a\neq -1$ and $a$ not a perfect square, the Artin density is $\delta(a)=\sum_{k\geq1}\mu(k)/[\mathbb Q(\zeta_k,a^{1/k}):\mathbb Q]$ [4] [1] [2].
Parameters
$a$
—   integer base ($a\neq -1$ and $a$ is not a perfect square)
Formulas
(1)
Let $h$ be the largest positive integer for which $a=a_0^h$ with $a_0\in\mathbb Z$, and let $d$ be the discriminant of $\mathbb Q(\sqrt a)$. With products over primes $q$, $A(h)=\prod_{q\nmid h}(1-\frac{1}{q(q-1)}) \prod_{q\mid h}(1-\frac{1}{q-1})$, and $\delta(a)=A(h)$ if $d\not\equiv1\pmod4$, while $\delta(a)=A(h)\left(1-\mu(|d|) \prod_{\substack{q\mid d\\q\mid h}}\frac{1}{q-2} \prod_{\substack{q\mid d\\q\nmid h}}\frac{1}{q^2-q-1}\right)$ if $d\equiv1\pmod4$ [2] [3].
(2)
Artin's constant is $A=A(1)=\delta(2)=\prod_q\left(1-\frac{1}{q(q-1)}\right)$, where $q$ runs over the primes [6] [5].
(3)
Under the generalized Riemann hypothesis for the fields $\mathbb Q(\zeta_k,a^{1/k})$ with $k$ squarefree, $\#\{p\leq x:a\text{ is a primitive root modulo }p\}\sim \delta(a)x/\log x$ [1] [2].
Comments
(4)
Perfect squares and $-1$ are omitted because their densities are $0$; perfect powers such as $8$ are included because their maximal exponent changes the rational factor in (1).
(5)
Unconditionally, Artin's primitive-root conjecture remains open for each fixed base $a$ in this table.
(6)
Every entry is an exact rational multiple of Artin's constant $A=A(1)=\delta(2)$ [6] [5]. The entry comment gives that multiple and the discriminant $d$ used in (1).
Programs
(P1)
Sage
from sage.libs.pari import pari
pari('default(realprecision, 140)')
A = pari('prodeulerrat(1 - 1/(p*(p - 1)))')
pari('20/19') * A           # a = 5
(P2)
PARI/GP
default(realprecision, 140)
A = prodeulerrat(1 - 1/(p*(p - 1)))
A                              \\ a = 2
(20/19) * A                    \\ a = 5
(3/5) * A                      \\ a = 8
References
[1]
C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209-220. (doi)
[2]
P. Moree, Artin's primitive root conjecture--a survey, Integers 12A (2012), A13. (arXiv)
[3]
P. Stevenhagen, The correction factor in Artin's primitive root conjecture, J. Théor. Nombres Bordeaux 15 (2003), no. 1, 383-391. (doi) (zbMATH) (MR)
Links
Similar tables
Values of the prime zeta function at rational numbers —   Artin's constant and the Euler products in (1) can be expanded through prime zeta values
Twin prime constant —   another named Euler product over primes defining a conjectural prime-density constant
Hardy-Littlewood singular series of prime tuples —   another table of constants in conjectural asymptotics for prime patterns
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds every integer base $a$ with $-50\leq a\leq 50$ satisfying the parameter constraint)
How they were obtained:

Artin's constant $A$ was computed with PARI's prodeulerrat at $140$ decimal digits, and $100$ digits are written. Each value is then an exact rational multiple of $A$, with the multiplier computed in $\mathbb Q$ from the maximal exponent $h$ and the discriminant $d$ in (1).

more

The value of $A$ agrees with OEIS A005596 [6] when rounded to $100$ decimal places, and recomputing it at $160$ decimal digits gives the same $100$ digits. The multipliers for all $92$ entries were checked against a second implementation of Hooley's formula. Among primes $p\leq 3\cdot10^6$, the observed fractions for $a=2,5,8,-3,21$, divided by $\pi(3\cdot10^6)$, differ from $\delta(a)$ by at most $0.135\%$.



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