A Riemann Hypothesis for characteristic p L-functions

Abstract

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute values. Further we use absolute values to give similar reformulations of the classical conjectures (with, perhaps, finitely many exceptional zeroes). We show how both sets of conjectures behave in remarkably similar ways.