Number Theory

TAA Lemma 1: Arithmetic Unitarity. The Riemann Hypothesis as Absence of Ghosts and Runaway Modes for the Scaling Flow of the Weil Form

Authors: J. W. McGreevy

On C_c^infty(R) (functions of the logarithmic scale u = log x), the Weil explicit-formula pairing Q is a Hermitian form, defined arithmetically by primes, the pole and the archimedean place, and spectrally by the zeros. The scaling flow T_t F(u) = F(u - t) is always Q-isometric (pseudo-unitary). We show: (a) the exponential growth rate of the matrix coefficients t -> Q(T_t F, G) is exactly Theta - 1/2, where Theta = sup Re rho; (b) RH <=> Q >= 0 <=> the flow is realized by a strongly continuous unitary group on a Hilbert space <=> all matrix coefficients are bounded; (c) the number of negative squares of Q equals the number of distinct zeros with Re rho > 1/2. Under RH the generator has simple pure point spectrum equal to the set of zero ordinates. Nothing here proves or disproves RH; parts (b) and (c) are in substance due to Weil and Bombieri, and the paper records exactly what is new. Part of the Thermodynamics of Arithmetic Action (TAA) program.

Comments: 6 Pages. TAA working notes.

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[v1] 2026-10-21 20:24:57

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