Digit permutations revisited

TL;DR

This paper explores the symmetries of characteristic p L-series using the group S_{(q)}, discusses natural actions and their extensions, and highlights the significance of unramified zeroes as an analogue of the Riemann hypothesis in finite characteristic.

Contribution

It introduces the group S_{(q)} as a symmetry group for characteristic p L-series and analyzes the uniqueness of extensions of its actions, especially when zeroes are unramified.

Findings

Unramified zeroes correspond to a unique extension of symmetries.

The group S_{(q)} acts naturally on characteristic p L-series.

Unramified zeroes are linked to an analogue of the Riemann hypothesis.

Abstract

We discuss here characteristic $p$ $L$-series as well as the group $S_{(q)}$ which appears to act as symmetries of these functions. We explain various actions of $S_{(q)}$ that arise naturally in the theory as well as extensions of these actions. In general such extensions appear to be highly arbitrary but in the case where the zeroes are unramified, the extension is unique (and it is reasonable to expect it is unique only in this case). Having unramified zeroes is the best one could hope for in finite characteristic and appears to be an avatar of the Riemann hypothesis in this setting; see Section [8] for a more detailed discussion.