Artin's Constant

Let n be a positive nonsquare integer. Then Artin conjectured that the set S(n) of all primes for which n 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 n be not an rth power for any r>1 such the squarefree part n^' of n satisfies n^'≢1 (mod 4). Let S^'(n) be the set of all primes for which such an n is a primitive root. Then Artin also conjectured that the density of S^'(n) relative to the primes is given independently of the choice of n by C_(Artin), where


(OEIS A005596), and p_k is the kth prime.

The significance of Artin's constant is more easily seen by describing it as the fraction of primes p for which 1/p has a maximal period repeating decimal, i.e., p is a full reptend prime (Conway and Guy 1996) corresponding to a cyclic number.

C_(Artin) is connected with the prime zeta function P(n) by


where L_n is a Lucas number (Ribenboim 1998, Gourdon and Sebah). Wrench (1961) gave 45 digits of C_(Artin), and Gourdon and Sebah give 60.

If n^'=1 (mod 4) and n is still restricted not to be an rth power, then the density is not C_(Artin) 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)

(Matthews 1976, Finch 2003), where mu(n) is the Möbius function. Special cases can be written down explicitly for n^'=p a prime,


or n^'=pq, where p,q are both primes with u,v=1 (mod 4),


If n 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

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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.", 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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Artin's Constant

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Weisstein, Eric W. "Artin's Constant." From MathWorld--A Wolfram Web Resource.

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