GAMMA FLOW

Written 5/10/25

Published 5/10/25

The Riemann zeta function, ζ(s), exhibits profound symmetries that are central to analytic number theory. The well-known functional equation relates ζ(s) to ζ(1 − s), suggesting a deep inversion symmetry around the critical line ℜ(s) = 1/2. However, this discrete symmetry does not fully account for the continuous structural transformations observed in the behavior of ζ(s) across vertical lines in the complex plane. This paper introduces a family of continuous affine transformations—termed “gamma flow”—that map the complex variable s while inducing corresponding transformations in the frequency spectrum of |ζ(s)|.

We define the gamma flow operator and an associated amplitude compensation operator that together establish what we call a“spectral quasi-symmetry” of the zeta function. This symmetry preserves the envelope of the Fourier spectrum under vertical line transformations when appropriate frequency scaling and amplitude corrections are applied. Our goal is to formalize this continuous transformation, present its analytic formulation, and provide numerical evidence for the preservation of spectral properties under the flow.