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Artin's Conjecture


There are at least two statements which go by the name of Artin's conjecture.

If r is any complex finite-dimensional representation of the absolute Galois group of a number field, then Artin showed how to associate an L-series L(s,r) with it. These L-series directly generalize zeta functions and Dirichlet L-series, and as a result of work by Richard Brauer, L(s,r) is known to extend to a meromorphic function on the complex plane. Artin's conjecture predicts that it is in fact holomorphic, i.e., has no poles, with the possible exception of a pole at s=1 (Artin 1923/1924). Compare with the generalized Riemann hypothesis, which deals with the locations of the zeros of certain L-series.

The second conjecture states that every integer not equal to -1 or a square number is a primitive root modulo p for infinitely many p and proposes a density for the set of such p which are always rational multiples of a constant known as Artin's constant. There is an analogous theorem for functions instead of numbers which has been proved by Billharz (Shanks 1993, p. 147).


See also

Artin's Constant, Artin L-Function, Generalized Riemann Hypothesis

Portions of this entry contributed by Mark Dickinson

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References

Artin, E. "Über eine neue Art von L-Reihen." Abh. Math. Sem. Univ. Hamburg 3, 89-108, 1923/1924.Matthews, K. R. "A Generalization of Artin's Conjecture for Primitive Roots." Acta Arith. 29, 113-146, 1976.Moree, P. "A Note on Artin's Conjecture." Simon Stevin 67, 255-257, 1993.Ram Murty, M. "Artin's Conjecture for Primitive Roots." Math. Intell. 10, 59-67, 1988.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 31, 80-83, and 147, 1993.

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Artin's Conjecture

Cite this as:

Dickinson, Mark and Weisstein, Eric W. "Artin's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ArtinsConjecture.html

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