Artin's Constant
Let 
be a positive nonsquare integer. Then
Artin conjectured that the set 
of all primes
for which 
is a primitive
root is infinite. Under the assumption of the generalized
Riemann hypothesis, Artin's conjecture was solved by Hooley (1967; Finch 2003,
p. 105).
Let 
be not an 
th power
for any 
such the squarefree
part 
of 
satisfies 
(mod 4). Let 
be the set
of all primes for which such an 
is a primitive
root. Then Artin also conjectured that the density of 
relative
to the primes is given independently of the choice
of 
by 
, where
![C_(Artin)=product_(k=1)^infty[1-1/(p_k(p_k-1))]=0.3739558136...](/images/equations/ArtinsConstant/NumberedEquation1.gif) |
(1)
|
(OEIS A005596), and 
is the 
th prime.
The significance of Artin's constant is more easily seen by describing it as the fraction of primes 
for which 
has a maximal period repeating
decimal, i.e., 
is a full
reptend prime (Conway and Guy 1996) corresponding to a cyclic
number.
is connected with the prime
zeta function 
by
 |
(2)
|
where 
is a Lucas
number (Ribenboim 1998, Gourdon and Sebah). Wrench (1961) gave 45 digits of 
, and Gourdon and Sebah give 60.
If 
and 
is still restricted
not to be an 
th power, then the density is not 
itself, but a rational multiple thereof.
The explicit formula for computing the density in this case is conjectured to be
![C_(Artin)^'=[1-mu(n^')product_(prime q; q|n^('))1/(q^2-q-1)]C_(Artin)](/images/equations/ArtinsConstant/NumberedEquation3.gif) |
(3)
|
(Matthews 1976, Finch 2003), where 
is the Möbius function. Special cases can be written
down explicitly for 
a prime,
 |
(4)
|
or 
, where 
are both primes with 
,
 |
(5)
|
If 
is a perfect cube (which is not a perfect
square), a perfect fifth power (which is not a perfect square or perfect cube), etc.,
other formulas apply (Hooley 1967, Western and Miller 1968).
SEE ALSO: Artin's Conjecture,
Barban's Constant,
Cyclic
Number,
Decimal Expansion,
Feller-Tornier
Constant,
Full Reptend Prime,
Heath-Brown-Moroz
Constant,
Murata's Constant,
Prime
Products,
Primitive Root,
Quadratic
Class Number Constant,
Sarnak's Constant,
Stephens' Constant,
Taniguchi's
Constant,
Twin Primes Constant
REFERENCES:
Artin, E. Collected Papers (Ed. S. Lang and J. T. Tate). New York: Springer-Verlag,
pp. viii-ix, 1965.
Conway, J. H. and Guy, R. K. The
Book of Numbers. New York: Springer-Verlag, p. 169, 1996.
Finch, S. R. "Artin's Constant." §2.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 104-110,
2003.
Gourdon, X. and Sebah, P. "Some Constants from Number Theory." https://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html.
Hooley, C. "On Artin's Conjecture." J. reine angew. Math. 225,
209-220, 1967.
Hooley, C. Applications of Sieve Methods to the Theory of Numbers. Cambridge, England: Cambridge
University Press, 1976.
Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag,
1990.
Lehmer, D. H. and Lehmer, E. "Heuristics Anyone?" In Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pólya
(Ed. G. Szegö, C. Loewner, S. Bergman, M. M. Schiffer,
J. Neyman, D. Gilbarg, and H. Solomon). Stanford, CA: Stanford University
Press, pp. 202-210, 1962.
Lenstra, H. W. Jr. "On Artin's Conjecture and Euclid's Algorithm in Global
Fields." Invent. Math. 42, 201-224, 1977.
Matthews, K. R. "A Generalization of Artin's Conjecture for Primitive Roots."
Acta Arith. 29, 113-146, 1976.
Ram Murty, M. "Artin's Conjecture for Primitive Roots." Math. Intell. 10,
59-67, 1988.
Ribenboim, P. The
New Book of Prime Number Records. New York: Springer-Verlag, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 80-83,
1993.
Sloane, N. J. A. Sequence A005596/M2608
in "The On-Line Encyclopedia of Integer Sequences."
Western, A. E. and Miller, J. C. P. Tables of Indices and Primitive Roots. Cambridge, England: Cambridge University
Press, pp. xxxvii-xlii, 1968.
Wrench, J. W. "Evaluation of Artin's Constant and the Twin Prime Constant."
Math. Comput. 15, 396-398, 1961.
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