$\zeta$-phenomenology

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.