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In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
Artin's conjecture on primitive roots; The (now proved) conjecture that finite fields are quasi-algebraically closed; see Chevalley–Warning theorem; The (now disproved) conjecture that any algebraic form over the p-adics of degree d in more than d 2 variables represents zero: that is, that all p-adic fields are C 2; see Ax–Kochen theorem or ...
Artin's conjecture on primitive roots states that a given integer a that is neither a perfect square nor −1 is a primitive root modulo infinitely many primes.
The Artin conjecture says Artin's L function is entire (holomorphic on the entire complex plane). automorphic form An automorphic form is a certain holomorphic function.
The Artin conjecture on Artin L-functions states that the Artin L-function (,) of a non-trivial irreducible representation ρ is analytic in the whole complex plane. [ 2 ] This is known for one-dimensional representations, the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions . [ 2 ]
Artin conjecture (L-functions) number theory: Emil Artin: 650 Artin's conjecture on primitive roots: number theory: ⇐generalized Riemann hypothesis [2] ⇐Selberg conjecture B [3] Emil Artin: 325 Bateman–Horn conjecture: number theory: Paul T. Bateman and Roger Horn: 245 Baum–Connes conjecture: operator K-theory: ⇒Gromov-Lawson ...
He left two conjectures, both known as Artin's conjecture. The first concerns Artin L-functions for a linear representation of a Galois group ; and the second the frequency with which a given integer a is a primitive root modulo primes p , when a is fixed and p varies.
Artin billiards; Artin braid group; Artin character; Artin conductor; Artin's conjecture for conjectures by Artin. These include; Artin's conjecture on primitive roots; Artin conjecture on L-functions; Artin group; Artin–Hasse exponential; Artin L-function; Artin reciprocity; Artin–Rees lemma; Artin representation; Artin–Schreier theorem ...