Riemann Hypothesis via Dynamical Stability and RG Flow

This attempts a conjectural proof of the Riemann Hypothesis (RH) based on the dynamical framework. Building on the Primal Manifold \(\mathcal{P}\), the kernel \(K(x,y)\), and the trace formula (see links below) to show that non-trivial zeros must lie on \(\Re(s) = 1/2\) as the only stable fixed point of the renormalization group (RG) flow. This relies on unproven assumptions from quantum chaos and RG theory (e.g., cyclic RG models). In section II below is a basic numerical verification of low-lying zeros and prime counts under RH as a refresher for students.

I. Conjectural Proof

Riemann Hypothesis

All non-trivial zeros of \(\zeta(s)\) have \(\Re(s) = 1/2\).

Draft Proof Idea

Stability Condition: From prior work (see links below), zeros are stable eigenstates where \(H_{\text{amp}} \psi = H_{\text{decay}} \psi\).
RG Flow: The beta function \(\beta = [H_{\text{amp}}, H_{\text{decay}}]\) has fixed point at \(\Re(s) = 1/2\), stable per linearization (negative eigenvalues).
Off-Line Instability: Disturbance away from this equilibrium, away from 1/2, then leads to divergence (where amplification dominates) or collapse (where decay dominates), making the zeros impossible.
Conclusion: Assuming renormalization group (RG) flow captures the scaling, RH holds.

II. Numerical Verification

Verify low-lying zeros by computing \(\zeta(1/2 + it) \approx 0\) for known \(t\), and prime count for \(x=1000\).

Zeta Zeros: For \(t \approx 14.1347\), \(\zeta(0.5 + it) \approx -1.05 \times 10^{-16} + 6.59 \times 10^{-16} i\); similarly for 21.0220 and 25.0109 — all near zero within precision.
Prime Count: For \(x=1000\), \(\pi(x) = 168\), \(\operatorname{li}(x) \approx 177.61\), \(\psi(x) \approx 996.68\); Riemann's formula with 100 zeros approximates \(\psi(x) \approx 993.97\), error \(2.71\) (improves with more zeros).