Comparison between the classical and characteristic p Riemann Hypotheses, by David Goss

This paper has now appeared under the title ``A Riemann hypothesis for characteristic p L-functions'' in J. Number Theory, vol. 82, no. 2, June 2000, 299-322, and so the preprint has been removed.

This is a new version of my paper with original abstract repeated below. In this new version I have corrected a couple of (minor) typo's. I have also worked very hard to make the paper accessible to non-experts. In particular, I have added a number of examples. The last example is significant in that it supports the conjectures and illustrates the importance of separability in the theory. This is done by creating an object with c.m. by inseparable elements; something not available in classical number theory. Original abstract (and also abstract for the second version): 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 formulations of the classical conjectures. We show how both sets of conjectures behave in remarkably similar ways.

David Goss <>