From: nikl+sm000315@pchelwig1.mathematik.tu-muenchen.de (Gerhard Niklasch)
Newsgroups: sci.math
Subject: Re: Artin Conjecture
Date: 20 Aug 1998 13:09:07 GMT

In article <35DB34F7.324B@watson.ibm.com>,
 Cliff <Zcliff@watson.ibm.com> writes:
|> Can anyone explain, in near-layperson terms, what the "Artin Conjecture"
|> is?   If that is not possible, can anyone explain what this is at
|> all?     It is said to be one of the most important unsolved
|> mathematical problems today.

There's more than one statement going by this name.

For a very readable introduction to the Artin Conjecture about primitive
roots, see H.W.Lenstra Jr., `Euclidean Fields' (3 parts), in The Mathe-
matical Intelligencer 2 #1 (1979), 6--15, 2 #2 (1980), 73--77 and 99--103.

This Artin Conjecture can be stated as saying that for a given positive
integer g, not a square, infinitely many primes p have the property
that g^k for 0 < k < p covers all the p-1 nonzero residue classes mod p.

(It can also be phrased in the rather more fancy language of class
field theory, and then becomes a statement about the splitting behaviour
of primes in abelian extensions of the field Q of rationals.)

This `conjeture' was proved by Hooley in 1967 assuming certain generalized
Riemann hypotheses.  So zeta functions (of number fields) have something
to do with the matter.

However, there is a far deeper Artin Conjecture involving zeta functions
of not necessarily abelian Galois extensions of Q and their decomposition
as a product of Riemann's zeta function and certain L functions attached
to representations of the Galois group involved (`Artin L functions').

One consequence of the general conjecture can be explained in terms of
zeros of zeta functions in the critical strip  (i.e., if all those
generalized Riemann hypotheses are true, on the line `Re z = 1/2');
namely, when K is a subfield of the field L which is Galois over Q
and when the function \zeta_K vanishes at the point s in the critical
strip, then so does \zeta_L  (and one can even say something about the
orders of vanishing).

It is quite feasible and quite amusing to check this experimentally
on the first several zeros, say for a field of degree 6, normal closure
of a non-cyclic cubic field, e.g. using PARI-GP.

Enjoy, Gerhard
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