Artin conjecture

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. Their Galois groups over the rational field Q are not abelian as soon as k > 2.

Using sieve methods, Roger Heath-Brown showed unconditionally that for all but at most two exceptional prime numbers q there are at least

cX/(logX)²

prime numbers p < X, such that q is a primitive root modulo p. This result is not constructive, as far as the exceptions go; it is of course conjectured that there are no exceptions.

See also: Brown-Zassenhaus conjecture .