Artin's conjecture, Turing's method and the Riemann hypothesis

TL;DR

This paper introduces a group-theoretic criterion to verify Artin's conjecture for certain Galois representations and explores methods for locating zeros of L-functions, with numerical tests supporting the conjectures.

Contribution

It provides a new criterion for verifying Artin's conjecture for non-monomial Galois representations and develops algorithms for analyzing zeros of L-functions.

Findings

Verification of Artin's conjecture for S5 and A5 representations.

Numerical evidence supporting the Riemann hypothesis for specific Dedekind zeta functions.

Development of algorithms for computing L-functions on the critical line.

Abstract

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S5 and A5 representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of non-abelian 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 jointly with Andreas Strombergsson) for computing with trace formulae. 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 S5 representations, and the Riemann hypothesis for Dedekind zeta functions of S5 and A5 fields.