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What is phenomenology?

It is well known that Euler experimentally discovered the functional equation of the Riemann zeta function. Indeed he detected the fundamental s → 1 − s invariance of ζ(s) by looking only at special values. In particular, via this functional equation, Z/(2) is realized as a group of symmetries of ζ(s). If one includes complex conjugation, then one has a group of symmetries of ζ(s) of order 4. 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 c = 2 ℵ0 (where c is the cardinality of the continuum); S (q) is a group of homeomorphisms of Z p which stabilizes both the nonpositive and nonnegative integers. We present three distinct instances in which S (q) acts (or appears to act) as symmetries of our functions; as of this writing, we do not yet know an overarching formalism that includes unifies these examples. Furthermore, we show that S (q) is naturally realized as an automorphism group of the convolution algebras of measures which appear in the theory. Finally we show how these ideas also naturally arise in the classical theory of Bernoulli numbers.