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In mathematics, there are two notable Artin conjectures, the legacy of Emil Artin.
The first of those concerns the region of the complex plane in which an Artin L-function is an analytic function. Let G be a Galois group of a finite Galois extension L/K of number fields; and let ρ be a group representation of G on a complex vector space of finite dimension. Then the Artin conjecture states that the Artin L-function
- L(ρ,s)
is meromorphic in the whole of the complex plane, having at most a pole at s = 1. Further, the multiplicity of the pole will be the multiplicity of the trivial representation in ρ.
This is known for one-dimensional representations — the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions. Further cases depend on the structure of G, when it is not an abelian group. Those seem to lie quite deep, for example in the work of Tunnell.
What is known in general comes out of Brauer's theorem on induced characters, which was in fact motivated by this application. It tells us, expressed in one kind of language, that the Q-module in the multiplicative group of non-zero meromorphic functions in the right half-plane Re(s) > 1 generated by the Hecke L-functions contains all the Artin L-functions. Here multiplication by 1/k means extraction of a k-th root of an analytic function; which is not a problem away from zeroes of the function, which we know do not occur in that half-plane. If there are zeroes, though, we may need branch cuts.
Therefore the Artin conjecture is concerned with zeroes of L-functions, just as the Riemann hypothesis family of conjectures is. It is believed that it would follow from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL(n) for all n ≥ 1. In fact this is a folk-theorem; it certainly represents one of the major motivations for the generality present in Langlands' work.
The second Artin conjecture relates to the density of the set of primes p modulo which a given integer a > 1 is a primitive root, when a is not a perfect square. For example, take a = 2. It claims that the set of primes p for which 2 is a primitive root has a density C, which is also (in fact) the heuristic 'probability of being a primitive root', namely the rate of growth of the sum
- Σ φ(p − 1)
summed over primes p up to X, and divided by X/logX to take an average. A more computable definition of C as an infinite product is given. Here φ(m) is Euler's totient function.
Hooley proved that the second conjecture is a consequence of the first (a conditional proof). He assumed the regularity of L-functions for certain extensions built by Kummer theory, by adjoining k-th roots of unity and the k-th root of a to the rational numbers.
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