$\zeta$-phenomenology

TL;DR

This paper explores a new symmetry group $S_{(q)}$ acting on characteristic $p$ zeta functions, revealing potential invariances and actions on zeros, extending classical symmetry concepts from Euler's work.

Contribution

It introduces the symmetry group $S_{(q)}$ for characteristic $p$ zeta functions and investigates its actions and potential unifying formalism.

Findings

$S_{(q)}$ acts as symmetries of characteristic $p$ zeta functions.

$S_{(q)}$ appears to stabilize zeta-zeroes.

$S_{(q)}$ can be realized as an automorphism group of convolution algebras.

Abstract

It is well known that Euler experimentally discovered the functional equation of the Riemann zeta function. Indeed he detected the fundamental $s\mapsto 1-s$ invariance of $\zeta(s)$ by looking only at special values. In particular, via this functional equation, the permutation group on two letters, $S_2\simeq\Z/(2)$, is realized as a group of symmetries of $\zeta(s)$. In this paper, we use the theory of special-values of our characteristic $p$ zeta functions to experimentally detect a natural symmetry group $S_{(q)}$ for these functions of cardinality ${\mathfrak c}=2^{\aleph_0}$ (where $\mathfrak c$ is the cardinality of the continuum); $S_{(q)}$ is a realization of the permutation group on $\{0,1,2...\}$ as homeomorphisms of $\Zp$ stabilizing both the nonpositive and nonnegative integers. We present a number of distinct instances in which $S_{(q)}$ acts (or appears to act) as symmetries of our functions. In particular, we present a natural, but highly mysterious, action of $S_{(q)}$ on a large subset of the domain of our functions that appears to stabilize zeta-zeroes. As of this writing, we do not yet know an overarching formalism that unifies these examples; however, it would seem that this formalism will involve an interplay between the 1-unit group $U_1$ -- playing the role of a "gauge group" -- and $S_{(q)}$. Furthermore, we show that $S_{(q)}$ may be naturally realized as an automorphism group of the convolution algebras of characteristic $p$ valued measures.