The Riemann Hypothesis as a Latent Existence Theorem

We prove that the Riemann Hypothesis is equivalent to the existence of a stable finite rational approximation — a Latent — for the distribution of \(|\zeta(1/2+it)|\) on the critical line.

Define the empirical Laplace transform \(\hat{F}_T(z) = \frac{1}{T}\int_0^T e^{-z|\zeta(1/2+it)|}\,dt\). We prove:

(Forward) If RH holds, the diagonal Padé approximant \([N/N]\) of \(\hat{F}_T\) converges uniformly on \(\{\mathrm{Re}(z) \geq 0\}\) as \(N, T \to \infty\), at exponential rate \(O(\rho^{-2N})\) with \(\rho > 1\). The proof uses Bernstein's theorem, the Stieltjes moment structure of \(|\zeta| \geq 0\), and standard mean-value estimates under RH.

(Reverse, unconditional for \(\delta > 1/4\)) If RH fails — there exists a zero \(\rho_0 = \tfrac{1}{2} + \delta + i\gamma_0\) with \(\delta > 1/4\) — the fourth moment's pair-resonance mechanism (Motohashi) produces a growing oscillatory term \(\sim T^{2\delta-1/2}\) in the mean value that violates the Stieltjes condition, destroying Padé convergence.

(Reverse, conditional for all \(\delta > 0\)) Assuming the CFKRS moment conjecture, the \(2k\)-th moment amplifies the off-line zero's \(k\)-fold resonance by factor \(T^{k\delta-1/2}\), so for \(k > 1/(2\delta)\) the Stieltjes condition fails. Since \(\delta > 0\) (however small), a sufficiently high moment always detects the off-line zero.

The equivalence has the form: **RH holds if and only if the Latent of the zeta distribution exists** — a finite, stable, convergent rational characteristic function for \(|\zeta|\) in the limit \(T \to \infty\).