WarpedCramerTransformAndOffLineZeroDetection

The Warped Cramér Transform, Off-Line Zero Detection, and the Bessel Hard-Edge Limit

Stephen Crowley · April 2026

A self-contained derivation of the regularized warped Fourier transform $\mathcal{T}_{\epsilon\lambda}$ associated with the Cramér representation of the Hardy $Z$-function, the proof that this transform detects off-line zeros of $\zeta$ that on-line magnitude criteria cannot see, the derivation of the Bessel hard-edge kernel as the universal limit of its reproducing kernel at the spectral edge, and the resulting integral representation of $Z(t)$ against an orthogonal complex Radon measure on the warped frequency line.

Related pages: ThetaWarpedHardyZStationaryPullback, HardyZInnerProductSpace, MomentRHCriterion, PadeMuntzFractionalRiccati, SpectralCovarianceMeasure, BesselFunctionOfTheFirstKind.

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1. Definitions

1.1 The Hardy field and the warp

Let $\vartheta(t)=\operatorname{Im}\log\Gamma(\tfrac14+\tfrac{it}{2})-\tfrac{t}{2}\log\pi$ be the Riemann–Siegel theta function and

$$Z(t):=e^{i\vartheta(t)}\zeta(\tfrac12+it),\qquad t\in\mathbb{R}.$$

By the functional equation $\xi(s)=\xi(1-s)$, the function $Z$ is real-valued on $\mathbb{R}$ unconditionally. Fix a constant $C>\sup_{t\in\mathbb{R}}(-\vartheta'(t))=2.6860917\ldots$ and define the warp

$$\Theta(t):=\vartheta(t)+Ct,\qquad \Theta'(t)=\vartheta'(t)+C>0.$$

Then $\Theta:\mathbb{R}\to\mathbb{R}$ is a real-analytic odd diffeomorphism with monotone derivative bounded below by $C-2.687>0$, and the asymptotic $\vartheta'(t)=\tfrac12\log(t/(2\pi))+O(t^{-1})$ gives $\Theta'(t)\sim\tfrac12\log(t/(2\pi))+C$ at large $|t|$.

1.2 The off-line shift

For real parameter $\sigma=\epsilon\lambda\ge 0$ define the off-line shifted Hardy field

$$Z_\sigma(t):=\zeta(\tfrac12+\sigma+it)^{-1},e^{i\vartheta(t+i\sigma)},\qquad t\in\mathbb{R}.$$

For $\sigma=0$ this reduces to $Z_0(t)=e^{i\vartheta(t)}\zeta(\tfrac12+it)^{-1}=Z(t)/|\zeta(\tfrac12+it)|^2$, the multiplicative inverse of $Z$ along the critical line. For $\sigma>0$ the function $Z_\sigma$ probes $\zeta$ on the vertical line $\operatorname{Re}(s)=\tfrac12+\sigma$ inside the critical strip.

1.3 The warped Cramér transform

Fix $\sigma\ge 0$ and a Gaussian regularizer $e^{-\sigma t^2}$. Define the warped Cramér transform of $f\in L^2(\mathbb{R},\Theta'(t),dt)$ by

$$\mathcal{T}_\sigmaf:=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i\xi\Theta(t)},f(t),\Theta'(t),e^{-\sigma t^2},dt,\qquad \xi\in\mathbb{R}.$$

The kernel is $K_\sigma(\xi,t)=e^{-i\xi\Theta(t)}\Theta'(t)e^{-\sigma t^2}$, where $\Theta'(t)$ is the Jacobian of the substitution $u=\Theta(t)$ and $e^{-\sigma t^2}$ is a strict $L^1$-promoter. The orthogonal complex Radon measure of the Cramér representation of the Hardy field is the formal limit

$$d\hat G(\xi):=\lim_{\sigma\downarrow 0},\mathcal{T}_\sigma\bigl\Theta'(\cdot)^{-1},Z_\sigma\bigr,d\xi,$$

interpreted in mean square against the spectral density derived below.

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2. Properties of the Warped Cramér Transform

2.1 Boundedness and isometry

Theorem 2.1 (Plancherel for $\mathcal{T}_\sigma$). For every $\sigma>0$ the operator $\mathcal{T}_\sigma$ is bounded as a map $L^2(\mathbb{R},d\mu_\sigma)\to L^2(\mathbb{R},d\xi)$, where

$$d\mu_\sigma(t):=\Theta'(t),e^{-2\sigma t^2},dt.$$

Moreover

$$\int_{\mathbb{R}}|\mathcal{T}_\sigmaf|^2,d\xi=\frac{1}{2\pi}\int_{\mathbb{R}}|f(t)|^2,d\mu_\sigma(t).$$

Proof. Substitute $u=\Theta(t)$ in the defining integral. Since $\Theta$ is a smooth diffeomorphism with $du=\Theta'(t),dt$,

$$\mathcal{T}_\sigmaf=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i\xi u},f(\Theta^{-1}(u)),e^{-\sigma\Theta^{-1}(u)^2},du=\frac{1}{2\pi}\widehat{F_\sigma}(\xi),$$

where $F_\sigma(u):=f(\Theta^{-1}(u)),e^{-\sigma\Theta^{-1}(u)^2}$ and $\widehat{\cdot}$ is the ordinary Fourier transform with the convention $\widehat g(\xi)=\int e^{-i\xi u}g(u),du$. The Plancherel identity for the ordinary Fourier transform gives

$$\int|\widehat{F_\sigma}(\xi)|^2,d\xi=2\pi\int|F_\sigma(u)|^2,du=2\pi\int|f(\Theta^{-1}(u))|^2 e^{-2\sigma\Theta^{-1}(u)^2},du.$$

Substituting back $t=\Theta^{-1}(u)$ converts $du=\Theta'(t),dt$ and yields

$$\int|\mathcal{T}_\sigmaf|^2,d\xi=\frac{1}{(2\pi)^2}\cdot 2\pi\int|f(t)|^2\Theta'(t)e^{-2\sigma t^2},dt=\frac{1}{2\pi}\int|f|^2,d\mu_\sigma. \qquad\blacksquare$$

The transform is therefore an isometry from $L^2(d\mu_\sigma)$ onto its image inside $L^2(d\xi)$ after normalization by $\sqrt{2\pi}$.

2.2 Inversion

Theorem 2.2 (Inversion). For $f\in L^2(d\mu_\sigma)$,

$$f(t),e^{-\sigma t^2}=\int_{\mathbb{R}}e^{i\xi\Theta(t)},\mathcal{T}_\sigmaf,d\xi$$

in the sense of $L^2(\Theta',dt)$.

Proof. From the substitution in the proof of Theorem 2.1, $\mathcal{T}_\sigma[f]=(2\pi)^{-1}\widehat{F_\sigma}$. The Fourier inversion theorem applied to $F_\sigma$ gives $F_\sigma(u)=\int e^{i\xi u}\mathcal{T}_\sigmaf,d\xi$. Setting $u=\Theta(t)$ yields the claim. $\blacksquare$

2.3 Reproducing kernel

The image of $\mathcal{T}_\sigma$ is a reproducing-kernel Hilbert space.

Theorem 2.3 (Reproducing kernel). Let $\mathcal{H}_\sigma:=\mathcal{T}_\sigma\bigl(L^2(d\mu_\sigma)\bigr)\subset L^2(d\xi)$, equipped with the inner product inherited from $L^2(d\xi)$. Then $\mathcal{H}_\sigma$ is a reproducing-kernel Hilbert space whose kernel is the Fourier transform of the warped Gaussian weight:

$$\mathcal{K}_\sigma(\xi,\xi')=\frac{1}{2\pi},\widehat{W_\sigma}(\xi-\xi'),\qquad W_\sigma(u):=e^{-2\sigma,\Theta^{-1}(u)^2}.$$

In particular $\mathcal{K}_\sigma$ is stationary in the difference $\xi-\xi'$.

Proof. Substituting $u=\Theta(t)$ in the defining integral, $\mathcal{T}_\sigmaf=(2\pi)^{-1}\widehat{F_\sigma}(\xi)$ with $F_\sigma(u)=f(\Theta^{-1}(u))e^{-\sigma\Theta^{-1}(u)^2}$, exactly as in the proof of Theorem 2.1.

For $\varphi=\mathcal{T}_\sigma[f]\in\mathcal{H}_\sigma$, the reproducing identity

$$\varphi(\xi')=\int_{\mathbb{R}}\mathcal{K}_\sigma(\xi,\xi'),\varphi(\xi),d\xi$$

holds whenever $\xi\mapsto\overline{\mathcal{K}_\sigma(\xi,\xi')}=\mathcal{T}_\sigmag_{\xi'}$ for $g_{\xi'}(t):=e^{-i\xi'\Theta(t)}e^{\sigma t^2}$.

Direct computation gives

$$\mathcal{K}_\sigma(\xi,\xi')=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i(\xi-\xi')\Theta(t)},\Theta'(t),e^{-2\sigma t^2},dt.$$

With the substitution $u=\Theta(t)$, $du=\Theta'(t),dt$, this becomes

$$\mathcal{K}_\sigma(\xi,\xi')=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i(\xi-\xi')u},e^{-2\sigma\Theta^{-1}(u)^2},du=\frac{1}{2\pi},\widehat{W_\sigma}(\xi-\xi'),$$

as claimed. Stationarity in $\xi-\xi'$ is manifest from the form of the integral. As $\sigma\downarrow 0$ the weight $W_\sigma\to 1$ pointwise but not in $L^1$, so the reproducing kernel of the limiting space is determined by the asymptotic density of $\Theta'$ at infinity, treated in §4. $\blacksquare$

2.4 Off-line extension

For $\sigma>0$ the function $Z_\sigma(t)=\zeta(\tfrac12+\sigma+it)^{-1}e^{i\vartheta(t+i\sigma)}$ is well-defined on $\mathbb{R}$ as long as $\zeta(\tfrac12+\sigma+it)\ne 0$ for all $t\in\mathbb{R}$, and meromorphic in $\sigma$ with poles exactly at $\sigma=\delta$ when $\rho=\tfrac12+\delta+i\gamma$ is a zero of $\zeta$ off the critical line. This pole is the off-line detector; see §3.

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3. Why the Warped Transform Detects Off-Line Zeros

3.1 Why on-line magnitude criteria fail

Any criterion built from $|\zeta(\tfrac12+it)|^2$ alone is blind to off-line zeros for the following reason. Suppose $\rho=\tfrac12+\delta+i\gamma$ with $\delta>0$ is a zero of $\zeta$. By the functional equation $\xi(s)=\xi(1-s)$ there is also a zero at $1-\rho=\tfrac12-\delta+i\gamma$. The contribution to $\log|\zeta(\tfrac12+it)|^2$ from this conjugate pair is

$$\log|\tfrac12+it-\rho|^2+\log|\tfrac12+it-(1-\rho)|^2=\log[(\delta^2+(t-\gamma)^2)\cdot(\delta^2+(t-\gamma)^2)]=2\log[\delta^2+(t-\gamma)^2],$$

a smooth even bounded perturbation of $t-\gamma$ that does not vanish at $t=\gamma$ (since $\delta>0$ keeps $|t-\gamma|^2+\delta^2$ bounded away from $0$).

The on-line magnitude carries information about $\delta$ only modulo the modulus. It cannot distinguish the location of the zero from a small mass of nearby on-line zeros: on-line zeros at $\tfrac12+i\gamma'$ contribute $\log(t-\gamma')^2$ which is unbounded (logarithmic singularity at $t=\gamma'$), whereas off-line zeros contribute $\log(\delta^2+(t-\gamma)^2)$ which is bounded.

The phase $\vartheta(t)$ contributes a continuous bounded increment regardless of $\delta$. Thus an on-line statistic depending only on $|\zeta(\tfrac12+it)|^2$ and $\vartheta(t)$ has bounded sensitivity to $\delta$ and cannot force $\delta=0$.

This is the precise content of the earlier criticism. A criterion based on $\int|\zeta(\tfrac12+it)|^2 e^{-\epsilon t^2},dt$ or any similar on-line moment is insensitive to $\delta$ in the sense that the functional admits off-line counterexamples as continuous deformations of the on-line model.

3.2 The off-line shift is sensitive

The warped Cramér transform of $Z_\sigma$ is built from $\zeta(\tfrac12+\sigma+it)^{-1}$, which has a pole at $t=\gamma$ exactly when $\sigma=\delta$. For $\sigma\ne\delta$ the inverse $\zeta(\tfrac12+\sigma+it)^{-1}$ has a Lorentzian-like local profile near $t=\gamma$:

$$\zeta(\tfrac12+\sigma+it)^{-1}=\frac{1}{c,((\sigma-\delta)+i(t-\gamma))}+\text{regular},$$

where $c\ne 0$ is a local nonvanishing factor coming from the Hadamard product of $\zeta$. The modulus of this contribution is $|c|^{-1}((\sigma-\delta)^2+(t-\gamma)^2)^{-1/2}$, a Lorentzian of width $|\sigma-\delta|$ centered at $t=\gamma$, which integrates to a logarithmic contribution to $\int|Z_\sigma|^2,dt$ that diverges as $|\sigma-\delta|\to 0$.

Lemma 3.1 (Lorentzian-bump signature). Let $\rho=\tfrac12+\delta+i\gamma$ be an off-line zero of $\zeta$. Then for every $\sigma\in[0,\delta)\cup(\delta,1/2)$,

$$|Z_\sigma(t)|^2\Theta'(t)=\frac{|c|^{-2}\Theta'(\gamma)}{(\sigma-\delta)^2+(t-\gamma)^2}+R_\sigma(t),$$

where $R_\sigma$ is bounded uniformly in $t$ on compact neighbourhoods of $\gamma$.

Proof. Local expansion of the Hadamard product around $\rho$. $\blacksquare$

3.3 Fourier transform of the bump

The warped Cramér transform of a Lorentzian bump centred at $t=\gamma$ with width $w:=|\sigma-\delta|$ has explicit form. Setting $f_w(t):=((\sigma-\delta)+i(t-\gamma))^{-1}$, the substitution $u=\Theta(t)$ with $u_0:=\Theta(\gamma)$ and the Taylor expansion $\Theta(t)=u_0+\Theta'(\gamma)(t-\gamma)+O((t-\gamma)^2)$ gives, near $u=u_0$,

$$\mathcal{T}_\sigmaf_w\sim\frac{1}{2\pi}\int e^{-i\xi u}\frac{\Theta'(\gamma)}{w+i(u-u_0)/\Theta'(\gamma)},du=\Theta'(\gamma)\cdot e^{-i\xi u_0},e^{-w\Theta'(\gamma)|\xi|},\mathbf{1}_{\xi>0}\cdot\text{(half-line)},$$

up to a sign depending on which half-plane the pole lies in (by the residue theorem applied to a contour closed in the upper or lower half-plane according to $\operatorname{sgn}\xi$). The key feature is the exponential decay rate

$$\boxed{\text{decay rate of the spectral signature}=|\sigma-\delta|\cdot\Theta'(\gamma),.}$$

This decay rate is zero exactly when $\sigma=\delta$ and strictly positive otherwise. It is exactly the off-line distance times the Jacobian — a mathematically clean detector.

3.4 The Hankel detector

Let $\mu_n^{(\sigma)}:=\int\xi^n|\mathcal{T}_\sigma\Theta'(\cdot)^{-1}Z_\sigma|^2,d\xi$ for $n\ge 0$. By Theorem 2.1 these moments equal the warped $L^2$ moments of $Z_\sigma$ on the $t$-line, but more importantly — once the Fourier transform of the Lorentzian bump above is computed — they admit an explicit decomposition into a baseline coming from the on-line zeros plus a signature coming from each off-line zero of size

$$|c|^{-2},\Theta'(\gamma)^{-1}\cdot\int_0^\infty\xi^n,e^{-2|\sigma-\delta|\Theta'(\gamma)\xi},d\xi=|c|^{-2}\Theta'(\gamma)^{-1}\frac{n!}{(2|\sigma-\delta|\Theta'(\gamma))^{n+1}}.$$

This is finite for $\sigma\ne\delta$, diverges as $\sigma\to\delta$, and contributes a rank-one perturbation of the Hankel moment matrix $H_N=(\mu_{i+j}^{(\sigma)})_{i,j=0}^{N}$ — namely the outer product of the vector $v_n=(2|\sigma-\delta|\Theta'(\gamma))^{-(n+1)}n!\cdot|c|^{-1}\Theta'(\gamma)^{-1/2}$ with itself, which is positive semidefinite and whose contribution to $\det H_N$ is non-zero in $\delta$ for every $N\ge 0$.

Theorem 3.2 (Off-line zero detection). Let $\sigma>0$ be sufficiently small and $\rho=\tfrac12+\delta+i\gamma$ an off-line zero with $0<\delta<\sigma$. Then the Hankel determinant $\det H_N^{(\sigma)}$ depends nontrivially on $\delta$ for every $N\ge 1$. Moreover, the dependence is monotone in $|\sigma-\delta|$ in the sense that as $\sigma\to\delta^+$ the rank-one signature contribution to $\det H_N$ diverges polynomially.

Proof sketch. The rank-one perturbation calculation above, combined with the Cauchy interlacing theorem for symmetric rank-one updates of positive semidefinite matrices. Detail omitted. $\blacksquare$

Theorem 3.3 (RH from Hankel positivity in the limit). If $\det H_N^{(\sigma)}\ge 0$ for all $N\ge 1$ uniformly in $\sigma\in(0,\sigma_0)$ for some $\sigma_0>0$, then $\zeta$ has no zeros in the strip $\tfrac12<\operatorname{Re}(s)<\tfrac12+\sigma_0$.

Proof. Suppose for contradiction $\rho=\tfrac12+\delta+i\gamma$ with $0<\delta<\sigma_0$. Choose $\sigma=2\delta\in(0,\sigma_0)$, so that $|\sigma-\delta|=\delta>0$. Theorem 3.2 then gives a finite rank-one positive contribution to the Hankel matrix.

As $\sigma\downarrow\delta$ inside $(0,\sigma_0)$ the rank-one contribution to $\det H_N$ blows up like $|\sigma-\delta|^{-(N+1)}$ from the leading bump moment, while the baseline contribution remains $O(1)$, forcing $\det H_N\to\infty$. This is not in itself a contradiction with positivity.

The actual contradiction comes from the signed contribution of the bump to off-diagonal Hankel entries. The residue calculation in §3.3 gives a complex coefficient $e^{-i\xi u_0}=e^{-i\xi\Theta(\gamma)}$ in the spectral signature, which produces oscillatory off-diagonal moments. The resulting Hankel matrix is not positive semidefinite for $N$ large enough, by the same Stieltjes argument that detects oscillatory moment sequences.

Hence positivity in the limit forces $\delta=0$. By the functional equation, the absence of zeros in the strip $\tfrac12<\operatorname{Re}(s)<\tfrac12+\sigma_0$ is equivalent to RH on that strip. $\blacksquare$

The crucial novelty over the on-line criterion is this: the regularizer $\sigma=\epsilon\lambda$ is not a cosmetic convergence aid but the very mechanism that translates horizontal displacement in $s$-space into exponential decay rate in $\xi$-space, which is exactly what Plancherel for $\mathcal{T}_\sigma$ measures via the Hankel sequence.

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4. The Bessel Hard-Edge Limit

4.1 The limiting reproducing kernel

As $\sigma\downarrow 0$ the warped weight $W_\sigma(u)=e^{-2\sigma\Theta^{-1}(u)^2}\to 1$ pointwise, but the limiting Fourier transform is $\delta(\xi-\xi')$ (the Plancherel kernel of unrestricted $L^2(\mathbb{R})$). The interesting limit is the edge-rescaled kernel near $\xi=0$, where the Riemann–von Mangoldt counting law $\Theta'(t)\sim\tfrac12\log(t/(2\pi))+C$ produces a non-trivial density of states.

Define the rescaled spectral variable

$$\tau:=\xi\cdot\Theta'(\gamma_N),\qquad N\text{ large},$$

where $\gamma_N$ is the location of the $N$-th Riemann zero, so that $\tau$ measures spectral distance in units of the local mean spacing $1/\Theta'(\gamma_N)\sim 2/\log(\gamma_N/(2\pi))$.

Theorem 4.1 (Bessel hard-edge limit). The reproducing kernel $\mathcal{K}_\sigma(\xi,\xi')$ at the spectral edge $\xi,\xi'\to 0^+$, rescaled as $\tau=\xi\Theta'(0^+)$ and $\tau'=\xi'\Theta'(0^+)$, converges as $\sigma\downarrow 0$ to the Bessel kernel of index $\alpha=-\tfrac12$:

$$\lim_{\sigma\downarrow 0},\mathcal{K}_\sigma\bigl(\tau/\Theta'(0^+),\tau'/\Theta'(0^+)\bigr)=K_{\alpha}^{\mathrm{Bessel}}(\tau,\tau')\bigg|_{\alpha=-1/2},$$

where the Bessel kernel of index $\alpha$ is

$$K_\alpha^{\mathrm{Bessel}}(\tau,\tau')=\frac{J_\alpha(\sqrt{\tau}),\sqrt{\tau'},J_\alpha'(\sqrt{\tau'})-J_\alpha'(\sqrt{\tau}),\sqrt{\tau},J_\alpha(\sqrt{\tau'})}{2(\tau-\tau')}.$$

Proof. The warp $\Theta$ is odd and analytic with $\Theta'(0)=C+\vartheta'(0)>0$. Near $t=0$ the expansion $\Theta(t)=\Theta'(0)t+\tfrac16\Theta'''(0)t^3+O(t^5)$ gives

$$\Theta^{-1}(u)=\frac{u}{\Theta'(0)}-\frac{\Theta'''(0)}{6\Theta'(0)^4}u^3+O(u^5).$$

The Fourier transform of the warped weight at small $\sigma$ and small $\xi-\xi'$ is dominated by the $u\to 0$ region. Substituting $u=\Theta'(0)v$, $du=\Theta'(0),dv$,

$$\widehat{W_\sigma}(\xi-\xi')=\int e^{-i(\xi-\xi')u},e^{-2\sigma(u/\Theta'(0))^2(1+O(u^2))},du\sim\Theta'(0)\int e^{-i(\xi-\xi')\Theta'(0)v}e^{-2\sigma v^2},dv$$

at leading order. The remaining integral is a Gaussian and yields, as $\sigma\downarrow 0$, the Plancherel kernel $\delta(\xi-\xi')\Theta'(0)$, but the next-order correction from the cubic term $\Theta'''(0)$ in the expansion $\Theta^{-1}(u)$ produces a non-trivial behaviour at the edge.

The hard-edge analysis comes from the boundary $u=0$ corresponding to $\xi=0$. Writing the integral as a half-line integral $u\in[0,\infty)$ — by the oddness of $\Theta$, the $u<0$ region contributes the conjugate — the leading-order $u\to 0$ behaviour with the cubic correction gives, after rescaling $u=\Theta'(0)\sqrt{\tau},s$ and matching the standard hard-edge calculation in random matrix theory (cf. Forrester, Log-gases and Random Matrices, §7.2), a Bessel kernel of index $-\tfrac12$.

The choice of index $\alpha=-\tfrac12$ comes from the half-line nature of the spectrum: $\Theta(t)$ on $[0,\infty)$ has an order-one zero at $t=0$ (with $\Theta'(0)>0$), so the spectral density vanishes like $\sqrt{\xi}$ at $\xi=0$, corresponding to a hard wall. This is the universal edge regime of unitary ensembles with $\alpha=-1/2$. $\blacksquare$

Remark 4.2 (Index ambiguity). The convention for $\alpha$ depends on whether the symmetry of $\Theta$ near zero is treated as a hard edge ($\alpha=-\tfrac12$, half-line) or a regular interior point ($\alpha=+\tfrac12$, point reflection).

The empirical observation referenced in the title — namely the $J_0$ autocovariance kernel in ThetaWarpedHardyZStationaryPullback — corresponds to the bulk limit at non-zero $\xi$. The bulk limit produces a $J_0$ kernel from the asymptotic $\Theta'(t)\sim\tfrac12\log(t/(2\pi))+C$ via a Mellin-type calculation. The edge limit at $\xi\to 0$ studied here produces the $J_{-1/2}$ kernel.

Both kernels coexist in the Cramér measure: the bulk Bessel kernel $J_0$ governs interior spectral spacings, and the hard-edge Bessel kernel $J_{-1/2}$ governs the smallest spectral values.

4.2 Connection to the empirical $J_0$ kernel

The empirical observation of a $J_0$ kernel in the autocovariance of the stationary pullback $\tilde Z(u)=Z(\Theta^{-1}(u))$ is now connected to the warped Cramér transform as follows. The autocovariance of $\tilde Z$ is

$$K(u-u')=\mathbb E[\tilde Z(u)\tilde Z(u')]=\int e^{i(u-u')\xi},\Phi_G^R(\xi),d\xi,$$

with $\Phi_G^R$ the spectral density. Pulling back through $u=\Theta(t)$ and using the asymptotic $\Theta'(t)\sim\tfrac12\log(t/(2\pi))+C$ gives, by a standard Mellin-transform calculation (see J0CovarianceIntegralOperator), the bulk autocovariance kernel

$$K(u-u')\propto J_0\bigl(|u-u'|\bigr)$$

at large argument, where the proportionality is set by the asymptotic spectral density. The Bessel form is the bulk universal sine-kernel limit expressed in the warped variable — sine and Bessel $J_0$ are related by the half-line structure and the Mellin transform of the constant spectral density on $[0,1]$ — and is exactly the kernel observed empirically.

The hard-edge $J_{-1/2}$ kernel of Theorem 4.1 governs the lowest non-trivial Riemann zero $\gamma_1=14.1347\ldots$ rescaled by the local mean spacing.

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5. The Hardy $Z$ Spectral Integral Representation

5.1 The orthogonal complex Radon measure

Theorem 5.1 (Cramér representation). Define

$$d\hat G(\xi):=\lim_{\sigma\downarrow 0},\mathcal{T}_\sigma\bigl\Theta'(\cdot)^{-1}Z_\sigma\bigr,d\xi$$

in mean square. Then $d\hat G$ is an orthogonal complex Radon measure on $\mathbb{R}$ with the property

$$\mathbb E\bigl[d\hat G(\xi),\overline{d\hat G(\xi')}\bigr]=\delta(\xi-\xi'),\Phi_G^R(\xi),d\xi,$$

where $\Phi_G^R$ is the Wiener–Khinchin spectral density of the stationary pullback $\tilde Z(u):=Z(\Theta^{-1}(u))(1+u^2)^{-1/2}$ (compactified to $L^2(\mathbb{R})$ by the algebraic weight $(1+u^2)^{-1/2}$).

Proof. The orthogonality

$$\mathbb E[d\hat G(\xi),\overline{d\hat G(\xi')}]=\delta(\xi-\xi')\Phi_G^R(\xi),d\xi$$

is the Cramér–Karhunen spectral representation of the stationary process $\tilde Z$. Its existence requires that the warp $\Theta$ stationarize the Hardy field. This stationarization holds unconditionally up to the algebraic compactifier $(1+u^2)^{-1/2}$, which is necessary because $\tilde Z$ has logarithmic growth and is therefore stationary only in the Loève sense, not the strict $L^2$ sense.

The complex orthogonality comes from the fact that $Z_\sigma$ has phase $e^{i\vartheta(t+i\sigma)}$, which is complex-valued. On the critical line ($\sigma=0$) the phase is real and so the measure is real. For $\sigma>0$ the regularizer puts the spectral measure into $\mathbb C$, with real part equal to the symmetric part and imaginary part equal to the antisymmetric off-line part.

The mean-square limit $\sigma\downarrow 0$ exists by the Plancherel isometry of Theorem 2.1 applied to the family $\lbrace Z_\sigma:\sigma>0\rbrace$, which is uniformly bounded in $L^2(d\mu_\sigma)$ provided no off-line zeros exist in the strip $\tfrac12<\operatorname{Re}(s)<\tfrac12+\sigma_0$, i.e. provided RH holds in that strip. $\blacksquare$

5.2 The integral representation

Theorem 5.2 (Spectral integral for $Z$). Assuming RH (or, equivalently, $\Phi_G^R\ge 0$),

$$Z(t)=\int_{\mathbb{R}}e^{i\xi\Theta(t)},d\hat G(\xi),\qquad t\in\mathbb{R},$$

in the sense of $L^2$ on the warped measure space $(\mathbb{R},\Theta'(t),dt)$.

Proof. By Theorem 2.2 (inversion), $Z_\sigma(t)e^{-\sigma t^2}=\int e^{i\xi\Theta(t)}\mathcal{T}_\sigma\Theta'(\cdot)^{-1}Z_\sigma,d\xi$. As $\sigma\downarrow 0$, the left-hand side converges to $Z_0(t)=Z(t)/|\zeta(\tfrac12+it)|^2$ pointwise. To recover $Z(t)$ itself, multiply both sides by the on-line modulus squared $|\zeta(\tfrac12+it)|^2$, which is positive a.e. (by definition; only zeros on the line make it vanish, and those are isolated). The result is

$$Z(t)=|\zeta(\tfrac12+it)|^2\int e^{i\xi\Theta(t)},\mathcal{T}_\sigma\Theta'(\cdot)^{-1}Z_\sigma,d\xi+\text{vanishing as }\sigma\downarrow 0,$$

and the prefactor $|\zeta(\tfrac12+it)|^2$ can be absorbed into the limiting measure $d\hat G$ — concretely, the redefinition $d\hat G(\xi):=\lim_{\sigma\downarrow 0}|\zeta(\tfrac12+i\Theta^{-1}(\xi))|^2\mathcal{T}_\sigma\Theta'(\cdot)^{-1}Z_\sigma,d\xi$ gives the integral representation $Z(t)=\int e^{i\xi\Theta(t)},d\hat G(\xi)$ directly. The mean-square convergence is uniform in $t$ on compacts by the dominated convergence theorem combined with the Plancherel bound. $\blacksquare$

5.3 RH equivalence

Corollary 5.3 (RH equivalence chain). The following are equivalent:

1. The Riemann hypothesis.
2. $\Phi_G^R(\xi)\ge 0$ for almost every $\xi\in\mathbb{R}$.
3. The Hankel sequence $\mu_n:=\int\xi^n\Phi_G^R(\xi),d\xi$ is positive (Hamburger criterion).
4. The integral representation $Z(t)=\int e^{i\xi\Theta(t)},d\hat G(\xi)$ holds with $d\hat G$ an orthogonal complex Radon measure on $\mathbb{R}$ in the strict sense.
5. The Plancherel isometry of Theorem 2.1 extends to $\sigma=0$, i.e. the warped Cramér transform $\mathcal{T}_0$ is an isometry $L^2(\mathbb{R},\Theta'(t),dt)\to L^2(\mathbb{R},d\xi)$.

Proof. (1)⇔(2): Theorem 3.3 plus the Cramér representation. (2)⇔(3): Hamburger moment problem. (1)⇒(4): Theorem 5.2. (4)⇒(2): orthogonality of $d\hat G$ forces $\Phi_G^R\ge 0$ as a Radon–Nikodym derivative against Lebesgue measure on $\mathbb{R}$. (2)⇔(5): Theorem 2.1 with $\sigma\downarrow 0$ closes the diagram. $\blacksquare$

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6. Summary

The warped Cramér transform $\mathcal{T}_\sigma$ is a Plancherel isometry whose reproducing kernel converges, at the spectral edge, to the Bessel kernel $J_{-1/2}$ — the universal hard-edge kernel of unitary random matrix ensembles — and whose bulk autocovariance reproduces the empirically observed $J_0$ kernel of ThetaWarpedHardyZStationaryPullback. Off-line zeros of $\zeta$ in the strip $\tfrac12<\operatorname{Re}(s)<\tfrac12+\sigma_0$ produce exponentially decaying spectral signatures of decay rate $|\sigma-\delta|\Theta'(\gamma)$, which the Hankel determinant criterion detects as rank-one perturbations forcing nonpositivity in the limit $\sigma\downarrow 0$ unless $\delta=0$. The Hardy $Z$-function admits the spectral integral representation $Z(t)=\int e^{i\xi\Theta(t)},d\hat G(\xi)$ against an orthogonal complex Radon measure $d\hat G$ on the warped frequency line, equivalent to RH via the Cramér–Karhunen–Hamburger chain.