Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

Abstract

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (nonmonomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to $S_5$ and $A_5$ representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of nonabelian extensions of $\Q$.

In addition, we discuss two methods for locating zeros of arbitrary $L$-functions. The first uses the explicit formula and techniques developed in A. R. Booker and A. Strömbergsson,“Numerical Computations with the Trace Formula and the Selberg Eigenvalue Conjecture,” for computing with trace formulas. The second method generalizes that of Turing for verifying the Riemann hypothesis. In order to apply it we develop a rigorous algorithm for computing general $L$-functions on the critical line via the fast Fourier transform.

Finally, we present some numerical results testing Artin's conjecture for $S_5$ representations, and the Riemann hypothesis for Dedekind zeta functions of $S_5$ and $A_5$ fields.