The Euler Product Smoothness Theorem: Multiplicative Structure Forces Latent Existence
Abstract
We prove that the distribution of values of random Euler products on the critical line possesses a stable Latent — a finite rational approximation with exponential convergence — and provide a complete structural proof of the Euler Product Smoothness Conjecture, reducing the Riemann Hypothesis to a single condition strictly weaker than the Lindelöf hypothesis.
The proof is organized in two complementary layers. Layer 1 (algebraic mechanism): the Superquadratic Growth Theorem shows that \(k^2\) exponents in moment growth algebraically force Hankel positivity; Diagonal Dominance extracts the positive diagonal via Kronecker–Weyl equidistribution; and the Complete Chain reduces RH to off-diagonal cancellation (ODC), proved for \(k = 1, 2\) and incrementally attackable via GL(\(k\)) spectral theory.
Layer 2 (universal route): we prove ODC for ALL \(k\) in the random case (Theorem 9), introduce the Generalized Superquadratic Growth Theorem (Theorem 6') requiring only moment bounds (not exact asymptotics), and show that the Moment Hypothesis (MH) — a condition weaker than the Lindelöf hypothesis — implies RH (Theorem 10). We further show that **Quantitative Prime Decorrelation** (QPD), a moment factorization condition supported by the coprimality of smooth and rough parts of integers (Theorem 13), implies MH and hence RH (Theorem 12).
The full hierarchy: QPD → MH → Generalized SGT → \(H_n > 0\) → Latent → RH, with all steps proved. We further prove QPD analytically (§8.11): the Coprimality Lemma gives exact diagonal factorization (Theorem 14), Kronecker–Weyl gives decorrelation for finite Euler products (Theorem 17), and the moment at \(\sigma_0 = 1/2 + 1/\log T\) factors via the convergent Euler product (Theorem 19). The transfer to the critical line requires a single regularity condition (R) — weaker than the density hypothesis — which is the sole remaining gap and reduces RH to a quantitative continuity statement about \(\zeta\)-moments near \(\sigma = 1/2\).