A Conditional Moment-Based Diagnostic for the Riemann Hypothesis

Tamás Nagy, Ph.D.

Working Paper — March 2026

Abstract

We propose a conditional moment-based diagnostic for the Riemann Hypothesis based on stable finite rational approximations — Latents — for the distribution of \(|\zeta(1/2+it)|\) on the critical line. The proposed equivalence and its converse remain open.

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

(Forward, conditional on a precise moment hypothesis) If RH holds together with the required higher-moment asymptotics, 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 \(\rho > 1\). This proposed route uses Bernstein's theorem, correctly signed Taylor coefficients, and additional moment assumptions not implied by RH alone.

(Reverse, conjectural 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 conjectural fourth-moment pair-resonance ansatz posits a growing oscillatory term \(\sim T^{2\delta-1/2}\) in the mean value. No established Motohashi theorem presently supplies this contribution or a converse to Padé convergence.

(Reverse, conditional proposal for all \(\delta > 0\)) Assuming the CFKRS moment conjecture, the \(2k\)-th moment amplifies the off-line zero's \(k\)-fold resonance by factor \(T^{k\delta-1/2}\) in the proposed ansatz. Establishing a property that detects this signal without contradicting the unconditional positivity of \(\mu_T\) remains open.

The intended criterion would have 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 \(T \to \infty\). This equivalence is a research objective, not a theorem proved here.

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1. Introduction

1.1 The Proposed Criterion

Let \(\zeta(s)\) denote the Riemann zeta function. For \(T > 0\), define the empirical distribution \(\mu_T\) of \(|\zeta(1/2+it)|\) on \([0,\infty)\):

\[\mu_T(A) = \frac{1}{T}\,\mathrm{meas}\{t \in [0,T] : |\zeta(\tfrac{1}{2}+it)| \in A\}\]

and its Laplace transform:

\[\hat{F}_T(z) = \int_0^\infty e^{-zx}\,d\mu_T(x), \quad \mathrm{Re}(z) > 0.\]

The Padé\([N\text{-}1/N]\) approximant \(R_N^T(z)\) is the unique rational function \(P_{N-1}(z)/Q_N(z)\) matching the first \(2N\) Taylor coefficients of \(\hat{F}_T\) at \(z = 0\).

Proposed Criterion 1. *The following conditions would be equivalent if the additional Padé and moment-resonance implications described below were established:*

(i) (RH) All non-trivial zeros of \(\zeta(s)\) satisfy \(\mathrm{Re}(s) = 1/2\).

(ii) (proposed stability condition) For every \(n \geq 0\), the Hankel determinant \(H_n(T) = \det[c_{i+j}(T)]_{i,j=0}^n\), where \(c_k(T) = m_k(T)/k!\) and \(m_k(T) = \int x^k\,d\mu_T(x)\), satisfies \(H_n(T) > 0\) for all sufficiently large \(T\), and converges: \(H_n(T) \to H_n^* > 0\).

(iii) (Padé convergence) The Padé\([N\text{-}1/N]\) approximant \(R_N^T\) converges uniformly on compact subsets of \(\{\mathrm{Re}(z) \geq 0\}\) as \(N, T \to \infty\).

(iv) (Latent existence) The Latent of \(\mu_T\) — the rational characteristic function \(\hat{\phi}_T(t) = R_N^T(it)\) — converges to a well-defined limit \(\hat{\phi}^*(t)\) as \(N, T \to \infty\).

Remark. The sequence \(m_k(T)\) is a Stieltjes moment sequence because \(\mu_T\) is positive; division by \(k!\) does not in general preserve that property. Thus (ii) is not currently justified by Bernstein's theorem, and neither the stated Padé implications nor a reverse implication from an off-line zero have been established. The proposed forward route requires a separate proof for the correctly signed Taylor-coefficient formulation; the proposed reverse route requires a moment-resonance theorem beyond current results.

1.2 Interpretation

The proposed criterion says:

> **The Riemann Hypothesis holds if and only if the primes are "smooth > enough" to have a finite rational description.**

In the Latent framework (Nagy, 2026e), a Latent is a finite, basis-free, sufficient representation of a smooth system. The Latent Theorem guarantees existence whenever the analyticity parameter \(\rho > 1\). Proposed Criterion 1 would connect this to RH if its required implications were proved. Whether an off-line zero forces the asserted change in \(\rho\), or prevents finite representation, is open.

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2. Preliminaries

2.1 The Stieltjes Moment Problem

Definition. A sequence \(\{c_k\}_{k \geq 0}\) of real numbers is a Stieltjes moment sequence if there exists a positive measure \(\sigma\) on \([0,\infty)\) such that \(c_k = \int_0^\infty x^k\,d\sigma(x)\).

Theorem (Stieltjes). \(\{c_k\}\) is a Stieltjes moment sequence if and only if the Hankel matrices \([c_{i+j}]_{i,j=0}^n\) and \([c_{i+j+1}]_{i,j=0}^n\) are positive semidefinite for all \(n \geq 0\).

2.2 Completely Monotone Functions

Theorem (Bernstein). \(f : (0,\infty) \to \mathbb{R}\) is completely monotone (i.e., \((-1)^k f^{(k)}(t) \geq 0\) for all \(k, t\)) if and only if \(f(t) = \int_0^\infty e^{-tx}\,d\mu(x)\) for some positive measure \(\mu\).

Corollary. If \(X \geq 0\) a.s., then \(g(z) = E[e^{-zX}]\) is completely monotone, and its correctly signed Taylor coefficients are \((-1)^k g^{(k)}(0)/k! = E[X^k]/k!\). Complete monotonicity does not by itself imply that the factorial-scaled sequence is a Stieltjes moment sequence.

Proof. \(g(z) = E[e^{-zX}]\) with \(X \geq 0\) is a Laplace transform of a positive measure. Bernstein's theorem gives complete monotonicity. The Stieltjes property applies directly to the unscaled moments \(E[X^k]\); any required property of the factorial-scaled coefficients must be proved separately. \(\square\)

2.3 Padé Approximants for Stieltjes Series

Theorem (Baker–Graves-Morris, §5.4). If \(\{c_k\}\) is a Stieltjes moment sequence, the diagonal Padé approximant \([N/N]\) of \(f(z) = \sum c_k z^k\) satisfies:

1. 1. All poles of \([N/N]\) lie on \((-\infty, 0)\) and interlace with the zeros.
2. 2. \([N/N]\) converges to the Stieltjes function

\(\int d\sigma(t)/(1-zt)\) on \(\mathbb{C} \setminus (-\infty, 0]\).

1. 3. The convergence is geometric: for the determinate case,

\(|f(z) - [N/N](z)| \leq C(z) \rho^{-2N}\) where \(\rho > 1\) depends on the support of \(\sigma\).

2.4 Mean Value Theorems for \(\zeta\)

Theorem (Ingham, 1926). For \(k = 1\): \[\int_0^T |\zeta(\tfrac{1}{2}+it)|^2\,dt = T\log\frac{T}{2\pi} + (2\gamma - 1)T + E_1(T)\] where \(E_1(T) = O(T^{1/2}\log T)\) unconditionally, and \(E_1(T) = O(T^{1/2+\varepsilon})\) under RH.

Theorem (Ingham, 1926). For \(k = 2\): \[\int_0^T |\zeta(\tfrac{1}{2}+it)|^4\,dt = T\,P_4(\log T) + E_2(T)\] where \(P_4\) is a specific degree-4 polynomial and \(E_2(T) = O(T^{2/3+\varepsilon})\) unconditionally.

Explicit formula for \(E_1\) (Titchmarsh, Ch. XV; Ivić, Ch. 4): \[E_1(T) = -2\,\mathrm{Re}\sum_\rho \frac{T^\rho}{\rho(1+\rho)} + O(\log T)\] where the sum runs over non-trivial zeros \(\rho\) of \(\zeta\).

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3. Conditional Forward Route: (i) \(\Rightarrow\) (ii)

Conditional Proposition 2. *Assume RH together with the stated higher-moment asymptotics and a separate factorial-scaled moment-sequence condition. Then the displayed normalized stability conclusion is the proposed forward route.*

Proof.

Step 1: Complete monotonicity at each \(T\).

For each \(T\), \(\hat{F}_T(z) = E_{\mu_T}[e^{-zX}]\) with \(X = |\zeta| \geq 0\). By Bernstein's theorem (§2.2), \(\hat{F}_T\) is completely monotone and the unscaled moments \(\{m_k(T)\}\) arise from a positive measure. This does not show that \(\{c_k(T)\}\) is a Stieltjes moment sequence; the needed factorial-scaled condition is an additional requirement.

Step 2: Moment convergence under RH.

Normalize: let \(\lambda_T = (\log T)^{1/2}\) and define the rescaled moments: \[\tilde{m}_k(T) = m_k(T) / \lambda_T^k = \frac{1}{T}\int_0^T \left(\frac{|\zeta(\tfrac{1}{2}+it)|}{\lambda_T}\right)^k dt\]

Under a precise Keating–Snaith-type moment conjecture, rather than from RH alone, the displayed structure is: \[\tilde{m}_{2k}(T) = \tilde{c}_k\,(\log T)^{k^2 - k} + O_k(T^{-1/2+\varepsilon})\]

for constants \(\tilde{c}_k\) depending on the Keating–Snaith coefficients. The displayed correction is an additional conjectural input. RH alone does not yield this bound for all \(k \geq 3\).

Step 3: Hankel determinant convergence.

Define \(\tilde{c}_k(T) = \tilde{m}_k(T)/k!\). Since \(\tilde{c}_k(T)\) converges for each \(k\) (to a limit that depends on \((\log T)^{k^2-k}\) — a polynomial in \(\log T\)), the Hankel determinant \(\tilde{H}_n(T) = \det[\tilde{c}_{i+j}(T)]\) is a continuous function of finitely many converging quantities, hence converges.

Under the additional factorial-scaled condition, the intended conclusion would follow by the stated continuity argument. Establishing strict positivity in this form is outside the present argument. \(\square\)

Remark. The Hankel determinants grow with \(\log T\) (because the moments grow). The displayed normalization motivates a conditional Padé diagnostic; it does not establish convergence or Latent existence.

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4. Conjectural Reverse Route: \(\neg\)(i) \(\Rightarrow\) \(\neg\)(ii)

4.1 The Second Moment (\(k = 1\)): Heuristic

Heuristic 3. *If there exists a zero \(\rho_0 = 1/2 + \delta + i\gamma_0\) with \(\delta > 1/2\), the displayed oscillatory calculation motivates a candidate diagnostic. It cannot imply sign changes of the Hankel determinant of the positive measure \(\mu_T\).*

Proof.

The explicit formula (§2.4) gives: \[m_2(T) = \log\frac{T}{2\pi} + (2\gamma-1) + E_1(T)/T\]

where \[E_1(T)/T = -\frac{2}{T}\,\mathrm{Re}\,\frac{T^{\rho_0}}{\rho_0(1+\rho_0)} + (\text{bounded terms})\]

\[= -\frac{2\,T^{\delta-1/2}\cos(\gamma_0\log T + \phi_0)}{|\rho_0(1+\rho_0)|} + O(T^{-1/2}\log T)\]

For \(\delta > 1/2\): \(T^{\delta - 1/2} \to \infty\), so \(m_2(T)\) oscillates with growing amplitude around \(\log(T/2\pi) + (2\gamma-1)\).

The normalized second moment \(\tilde{c}_1(T) = m_2(T)/(2\lambda_T^2)\) oscillates: for infinitely many \(T\), \(\tilde{c}_1(T) > \tilde{c}_1^{(\text{smooth})}\), and for infinitely many others, \(\tilde{c}_1(T) < \tilde{c}_1^{(\text{smooth})}\).

The displayed moment behavior does not imply that the Hankel determinant \(H_1(T) = c_0 c_2 - c_1^2\) becomes negative: positivity of \(\mu_T\) prevents such a conclusion. A valid reverse route must instead identify a property not automatically implied by positivity. \(\square\)

4.2 The Fourth Moment (\(k = 2\)): Conjectural Resonance Ansatz

Conjectural Ansatz 4. *If there exists a zero $\rho_0 = 1/2 + \delta + i\gamma_0\( with \)\delta > 1/4$, the displayed resonance calculation suggests a possible moment diagnostic. It does not establish failure of the Stieltjes condition.*

Proof.

We work with the Stieltjes moment problem for \(Y = |\zeta(1/2+it)|^2 \geq 0\), whose moments are \(\nu_k(T) = m_{2k}(T)\). The Hankel determinant: \[H_1^{(Y)} = \nu_0 \nu_2 - \nu_1^2 = m_4(T) - m_2(T)^2\]

Step 1: The smooth structure.

Under RH, \(m_2(T) \sim \log(T/2\pi)\) and \(m_4(T) \sim c_2\,(\log T)^4\). The smooth Hankel determinant: \[H_1^{(Y),\text{smooth}} \sim c_2\,(\log T)^4 - (\log T)^2 \sim c_2\,(\log T)^4\] which is positive and growing. Under RH, the perturbations decay as \(O(T^{-1/2+\varepsilon})\) and never dominate.

Step 2: The fourth moment pair resonance.

The fourth moment \(|\zeta(1/2+it)|^4 = |\zeta^2(1/2+it)|^2\) involves the mean square of \(\zeta^2\). The following is a conjectural resonance ansatz, not an established consequence of Motohashi's spectral theory. It supposes that the error \(E_2(T)\) contains contributions from pair resonances of ζ zeros. For a zero at \(\rho_0 = 1/2 + \delta + i\gamma_0\), the pair resonance contributes:

\[E_2^{(\rho_0)}(T) \sim C_2\, T^{2\rho_0 - 1/2}\, \cos(\gamma_0' \log T + \phi) = C_2\, T^{1/2+2\delta}\, \cos(\gamma_0' \log T + \phi)\]

The exponent \(2\rho_0 - 1/2 = 1/2 + 2\delta\) arises because the fourth moment captures the pair interaction of the zero with its functional-equation partner: the product \(\zeta(s)\zeta(1-\bar{s})\) at \(s = \rho_0\) generates a resonance at \(\rho_0 + \bar{\rho}_0 - 1/2 = 1/2 + 2\delta\).

Step 3: The mean value perturbation dominates.

The perturbation to the mean value: \[\Delta m_4(T) = E_2^{(\rho_0)}(T)/T \sim C_2\, T^{2\delta - 1/2}\, \cos(\cdots)\]

For \(\delta > 1/4\): \(2\delta - 1/2 > 0\), so \(|\Delta m_4|\) grows with \(T\).

The perturbation to \(H_1^{(Y)}\): \[\Delta H_1^{(Y)} \approx \Delta m_4 - 2\,m_2\,\Delta m_2\] The \(\Delta m_4 \sim T^{2\delta-1/2}\) term dominates (the \(\Delta m_2\) term is \(O(T^{\delta-1/2})\), which decays for \(\delta < 1/2\)).

The ratio: \[\frac{|\Delta H_1^{(Y)}|}{H_1^{(Y),\text{smooth}}} \sim \frac{T^{2\delta - 1/2}}{c_2\,(\log T)^4} \to \infty \quad \text{since } 2\delta - 1/2 > 0\]

The displayed oscillation cannot imply that \(H_1^{(Y)}(T)\) changes sign: \(Y = |\zeta(1/2+it)|^2\) induces a positive measure for every \(T\). The calculation instead identifies the type of non-positivity-based diagnostic that would need to be formulated.

Step 4: The threshold \(T_0(\delta)\).

The crossing occurs when \(|\Delta m_4(T_0)| \approx c_2\,(\ln T_0)^4\): \[T_0(\delta) \approx \exp\!\left(\frac{4\ln\ln T_0}{2\delta - 1/2}\right)\] For \(\delta = 0.30\): \(T_0 \approx e^{165} \approx 10^{72}\). For \(\delta = 0.48\): \(T_0 \approx e^{21} \approx 10^{9}\). For \(\delta \to 1/4^+\): \(T_0 \to \infty\) (the failure is asymptotic). \(\square\)

Remark. The pair-resonance formula \(E_2^{(\rho_0)} \sim T^{1/2+2\delta}\) is a conjectural ansatz. No established Motohashi theorem presently derives this off-line-zero contribution to the fourth-moment error.

4.3 Higher Moments: The Moment Amplification Principle

Conditional Ansatz 5 (CFKRS plus a resonance hypothesis). *If there exists a zero \(\rho_0 = 1/2 + \delta + i\gamma_0\) with any \(\delta > 0\), then for \(k > 1/(2\delta)\), the \(2k\)-th moment error satisfies:*

\[E_k(T) = \Omega(T^{1/2+k\delta})\]

*The mean value perturbation \(\Delta m_{2k}(T) \sim T^{k\delta-1/2}\) grows in the proposed ansatz. It cannot imply failure of a Hankel positivity condition that follows from the positive measure.*

Proof (conditional).

The CFKRS conjecture (Conrey, Farmer, Keating, Rubinstein, Snaith, 2005) motivates the following full asymptotic expansion: \[\int_0^T |\zeta(\tfrac{1}{2}+it)|^{2k}\,dt = T\sum_{j=0}^{k^2} a_{k,j}(\log T)^j + E_k(T)\]

The proposed resonance hypothesis asserts a contribution from each zero \(\rho\) through \(k\)-fold resonances in the autocorrelation of \(\zeta^k\):

1. 1. \(|\zeta(s)|^{2k} = |\zeta^k(s)|^2\) involves the mean square of \(\zeta^k\).
2. 2. The \(k\)-fold resonance of \(\rho_0\) produces a contribution at

$k\rho_0 + k\bar{\rho}_0 - (2k-1)/2 = 1/2 + 2k\delta - (k-1/2) = 1/2 + k\delta$ (after the Rankin–Selberg unfolding).

1. 3. The contribution to the integral: \(E_k^{(\rho_0)} \sim T^{1/2 + k\delta}\).

The mean value perturbation: \[\Delta m_{2k}(T) = E_k(T)/T \sim T^{k\delta - 1/2}\,\cos(\cdots)\]

For \(k > 1/(2\delta)\): \(k\delta > 1/2\), so \(T^{k\delta - 1/2} \to \infty\). The Hankel determinant at order \(n = \lfloor k/2 \rfloor\) involves \(\nu_0, \ldots, \nu_{2n}\) (i.e., \(m_0, m_2, \ldots, m_{4n}\)). The perturbation at the highest-order moment dominates the displayed smooth part. It does not force sign oscillation in a Hankel determinant of a positive measure. \(\square\)

The moment amplification ladder. Each moment detects progressively smaller deviations from the critical line:

\(k\)	Moment	Threshold \(\delta\)	\(\mathrm{Re}(\rho)\)	Status
1	\(m_2\)	\(1/2\)	\(1.00\)	Vacuous (no zeros with \(\text{Re} > 1\))
2	\(m_4\)	\(1/4\)	\(3/4\)	Conjectural resonance ansatz
3	\(m_6\)	\(1/6\)	\(2/3\)	Conditional on CFKRS (\(k=3\))
4	\(m_8\)	\(1/8\)	\(5/8\)	Conditional on CFKRS (\(k=4\))
\(k\)	\(m_{2k}\)	\(1/(2k)\)	\(1/2+1/(2k)\)	Conditional for \(k \geq 3\)

As \(k \to \infty\), the proposed threshold tends to \(\delta \to 0\); whether this yields a valid diagnostic for all off-line zeros remains open.

4.4 The Key Structural Insight

The preceding conditional and conjectural calculations motivate the *moment amplification principle*:

> An off-line zero with shift \(\delta\) is invisible to the \(2k\)-th moment > when \(k\delta < 1/2\), and visible when \(k\delta > 1/2\).

This suggests that sufficiently high moments may carry information about small shifts. The Padé construction uses all moments simultaneously, but its ability to detect every off-line zero is not proved; Padé convergence is not shown equivalent to RH or shown to fail under \(\neg\mathrm{RH}\).

In Latent language, this suggests a possible tail-sensitive diagnostic. Ordinary Hankel positivity cannot supply the reverse implication, because it already follows from the positivity of \(\mu_T\).

4.5 Illustrative Formal Computation Under the Ansatz

We record the formal sign-change calculation for \(H_1^{(Y)} = m_4 - m_2^2\) under the conjectural resonance ansatz, with a zero at \(\rho_0 = 1/2+\delta+i\gamma_0\), \(\delta > 1/4\). It is not a valid conclusion about the actual Hankel determinant.

The moments: \[m_2(T) = \log\frac{T}{2\pi} + (2\gamma-1) + A_1\,T^{\delta-1/2}\cos\alpha_1 + O(T^{-1/2}\log T)\] \[m_4(T) = c_2\,(\log T)^4 + (\text{lower}) + A_2\,T^{2\delta-1/2}\cos\alpha_2 + O(T^{\delta-1/2}\log T)\]

where \(\alpha_j = \gamma_j \log T + \phi_j\) are oscillatory phases.

The Hankel determinant: \[H_1^{(Y)} = m_4 - m_2^2 = \underbrace{[c_2(\log T)^4 - (\log T)^2 + \cdots]}_{\text{smooth, positive, } \sim c_2(\log T)^4} + \underbrace{A_2\,T^{2\delta-1/2}\cos\alpha_2}_{\text{dominant perturbation}} + O((\log T)\,T^{\delta-1/2})\]

The perturbation-to-smooth ratio: \[R(T) = \frac{|A_2|\,T^{2\delta-1/2}}{c_2\,(\log T)^4}\]

\[\log_{10} R(T) \approx (2\delta - \tfrac{1}{2})\,\log_{10} T - 4\,\log_{10}\log T + \text{const}\]

The slope in \(\log T\) is \((2\delta - 1/2)\log_{10} e \approx 0.434(2\delta - 1/2)\).

• For \(\delta > 1/4\): slope \(> 0\), so the formal ratio \(R(T) \to \infty\).

This is a feature of the ansatz, not evidence that \(H_1^{(Y)}\) changes sign or that the Stieltjes condition fails.

• For \(\delta = 1/4\): slope \(= 0\), boundary case (logarithmic competition).
• For \(\delta < 1/4\): slope \(< 0\), so \(R(T) \to 0\). The smooth part dominates.

The Stieltjes condition holds (at this Hankel order).

Numerical verification (see hankel_experiment.py):

\(\delta\)	\(\mathrm{Re}(\rho)\)	\(2\delta - 1/2\)	\(\log_{10} T_0\)	Status
0.255	0.755	0.01	1290	Formal ansatz scale
0.30	0.80	0.10	72	Formal ansatz scale
0.35	0.85	0.20	30	Formal ansatz scale
0.45	0.95	0.40	11	Formal ansatz scale
0.48	0.98	0.46	9.2	Formal ansatz scale

The formal threshold \(T_0(\delta) \to \infty\) as \(\delta \to 1/4^+\). No actual failure is established for any \(\delta > 1/4\).

4.6 Open Route: The Moment Bootstrap

The open gap in this proposed program concerns establishing a valid non-positivity-based diagnostic. The required explicit off-line-zero contributions to higher moment errors have not been established; CFKRS motivates an expected asymptotic but does not provide a rigorous proof.

Why the gap is hard. The pair-resonance mechanism (§4.2) that gives the \(T^{1/2+2\delta}\) contribution to \(E_2\) works because Motohashi's spectral decomposition captures the symmetric-square \(L\)-function of \(\zeta^2\), which naturally produces pair terms. For \(k = 3\), one would need the spectral theory of \(\zeta^3\), which involves \(\mathrm{GL}(3)\) automorphic forms — a theory that is not yet complete enough to extract the explicit zero contributions.

Three plausible routes to unconditional:

1. 1. GL(3) spectral theory. Extending Motohashi's method to \(k = 3\)

using the \(\mathrm{GL}(3)\) Voronoi formula (Blomer–Khan–Young, 2023+) would give the sixth moment explicit formula and establish the \(T^{1/2+3\delta}\) contribution, extending the range to \(\delta > 1/6\).

1. 2. Moment inequalities (bootstrap). If \(m_4(T)\) has a growing

perturbation, the log-convexity of moments (\(m_{2k}^2 \leq m_{2(k-1)} m_{2(k+1)}\)) constrains \(m_6\). When \(m_4\) oscillates upward, \(m_6 \geq m_4^2/m_2\) is forced above its smooth baseline. However, this gives a one-directional bound (lower, not upper), and proving bidirectional oscillation of \(m_6\) from \(m_4\) oscillation alone remains open.

1. 3. Harper's sharp moment bounds. Harper (2013) proved unconditional upper

bounds \(\int |\zeta|^{2k} \leq C_k T(\log T)^{k^2}\) for \(k \leq (\log\log T)^{1-\varepsilon}\). If an off-line zero with \(\delta > 0\) makes the \(k\)-fold resonance contribute \(\Omega(T^{1/2+k\delta})\) to the integral, this would eventually violate Harper's bound for \(k > (1/2)/(k\delta - 1/2)\) — but establishing the lower bound on the resonance contribution is precisely the CFKRS gap.

What is established today:

• Positivity of the moment Hankel matrices induced directly by \(\mu_T\).
• Classical low-moment estimates stated in §2.4.
• No forward equivalence, reverse implication, or Padé characterization of RH

claimed in this draft.

The Padé construction suggests a possible diagnostic because it uses all moments. The central open problem is to identify and prove a property that carries the required zero information without relying on a Hankel sign-change argument contradicted by positivity.

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5. The Equivalence in Latent Language

5.1 The Latent Framework

The Latent Theorem (Nagy, 2026e) states: every system with analyticity parameter \(\rho > 1\) has a finite sufficient representation (a Latent) of size \(N = \Theta(\log(1/\varepsilon)/\log\rho)\).

For the distribution of \(|\zeta|\):

• The Latent is the Padé rational characteristic function
• The analyticity parameter \(\rho\) is the Padé convergence rate
• The Latent would exist when \(\rho > 1\) under the separate Padé hypotheses

5.2 Proposed RH–Latent Relation

Proposed consequence of Criterion 1. *If the additional Padé and moment-resonance implications were established, RH would be equivalent to the existence of a well-defined Latent for the distribution of \(|\zeta(1/2+it)|\) in the limit \(T \to \infty\).*

Discussion. The asserted equivalences are not established in this draft. Condition (iv) describes the intended Latent conclusion if the proposed criterion can be proved. \(\square\)

5.3 The Two-Latent Decomposition

The moment function decomposes: \[m_{2k}(T) = \underbrace{P_k(\log T)}_{\text{smooth Latent }\mathcal{L}_{\text{smooth}}} + \underbrace{E_k(T)/T}_{\text{oscillatory Latent }\mathcal{L}_{\text{osc}}}\]

Under the additional moment assumptions: \(\mathcal{L}_{\text{osc}}\) is postulated to be bounded (\(O(T^{-1/2+\varepsilon})\)), which motivates a finite-dimensional model.

Under the conjectural resonance ansatz: \(\mathcal{L}_{\text{osc}}\) contains growing terms, suggesting a possible obstruction to such a model.

The unified Latent \(\mathcal{L} = \mathcal{L}_{\text{smooth}} \oplus \mathcal{L}_{\text{osc}}\) is a proposed model whose existence would require both components to be finite-dimensional.

The proposed interpretation is that RH could act as a closure condition for this Latent algebra. Neither direction of that interpretation is established.

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6. Numerical Evidence

6.1 CDF Accuracy

The Padé–Stieltjes machine recovers the CDF of \(|\zeta(1/2+it)|\) to 0.45% max error — 100× more accurate than the Selberg model (Nagy, 2026h). This demonstrates finite-sample approximation behavior at practical \(T\) values (\(T = 1000\)); it does not establish Latent existence in the limiting sense.

6.2 ζ-Zero Detection in Moment Residuals

Fourier analysis of \(E_k(T)/\sqrt{T}\) at \(T\) up to \(2 \times 10^6\) detects 10/15 of the first ζ-zero frequencies \(\gamma_n\) in the \(k = 2\) and \(k = 3\) residuals, at \(> 3\,\text{dB}\) signal-to-noise ratio. The strongest detection: \(\gamma_2 = 21.022\) at 8.2 dB.

This is numerically consistent with structured zero-frequency content in the residuals. It does not confirm the proposed theorem mechanism or establish high-moment amplification.

6.3 Padé Stability at Practical \(T\)

At all tested \(T\) values (\(10^3\) to \(2 \times 10^6\)), the Padé[6/7] poles lie on the negative real axis and the Hankel determinants are positive. This is consistent with positivity at the tested finite values; it does not verify the proposed forward implication.

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7. Discussion

7.1 What Is New

1. 1. A proposed conditional diagnostic for RH: the manuscript formulates a

possible connection between stable Padé approximation and RH. It is not an established equivalence. Previous Padé/moment characterizations (Li's criterion, Nyman–Beurling) are related but structurally different.

1. 2. The moment amplification proposal: the conditional resonance ansatz

identifies a possible way higher moments could carry off-line-zero information. Its diagnostic implication remains unproved.

1. 3. Computable invariants: the Hankel determinants \(H_n(T)\) and the Padé

convergence rate \(\rho_T\) are computable from moment data. This opens the possibility of numerical approaches to RH through Padé stability monitoring.

7.2 Relation to Li's Criterion

Li (1997) proved: RH \(\Leftrightarrow\) \(\lambda_n \geq 0\) for all \(n \geq 1\), where \(\lambda_n = \sum_\rho [1 - (1 - 1/\rho)^n]\). Our criterion is structurally similar — both involve positivity conditions on sequences derived from ζ — but ours operates on the MOMENTS of \(|\zeta|\) rather than the ZEROS directly. The moment-based formulation connects to approximation theory (Padé) and provides a different computational route.

7.3 Limitations

• The proposed reverse direction is not established for any \(\delta > 0\).
• Positivity of \(\mu_T\) rules out the claimed Hankel-sign mechanism for every

\(\delta\), including \(\delta > 1/4\).

• The requisite higher-moment resonance contributions are conjectural and are

not consequences of RH or current Motohashi theory.

• The displayed \(T_0(\delta)\) is a formal scale inside the conjectural ansatz,

not a mathematically guaranteed threshold.

7.4 The Open Challenge

To turn the proposed criterion into a theorem, one needs a valid non-positivity-based diagnostic and a rigorously established \(k\)-fold resonance contribution:

Conjecture (Unconditional Moment Amplification). For every \(k \geq 1\): \[\int_0^T |\zeta(\tfrac{1}{2}+it)|^{2k}\,dt = T\,P_k(\log T) + E_k(T)\] *where \(E_k(T)\) receives a \(k\)-fold resonance contribution from each zero \(\rho_0 = 1/2+\delta+i\gamma_0\) with \(\delta > 0\):* \[E_k^{(\rho_0)}(T) \sim C_k(\rho_0)\, T^{1/2+k\delta}\, \cos(\gamma_k\log T + \phi_k)\]

This is a conjectural ansatz motivated by ratios-conjecture and random-matrix heuristics. The classical \(k = 1\) explicit formula does not establish the claimed diagnostic, and no current Motohashi theorem establishes the stated \(k = 2\) off-line-zero contribution.

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8. Conclusion

The Riemann Hypothesis — the most important unsolved problem in mathematics — motivates the search for a finite rational representation (Latent) for the distribution of \(|\zeta(1/2+it)|\).

What this draft establishes and leaves open:

• Positive-measure structure: Bernstein's theorem supplies complete

monotonicity and unscaled moment positivity. It does not prove the factorial-scaled Stieltjes condition, Padé convergence, or Latent existence.

• Fourth-moment route: the pair-resonance expression

\(\sim T^{2\delta-1/2}\) is a conjectural ansatz, not an unconditional Motohashi consequence. It cannot prove Hankel-sign failure for the positive empirical measure.

• Higher-moment route: CFKRS and the stated resonance ansatz motivate a

possible amplification program, but no converse to RH has been proved.

The interpretation: the manuscript proposes that RH may be connected to whether the prime distribution is sufficiently regular to admit a stable finite representation. Establishing that connection requires a new diagnostic property not already forced by positivity of the empirical measure.

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During the preparation of this work the author used large language models in order to assist with manuscript drafting, literature search, and coding assistance. After using these tools, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article.

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