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WarpedCramerTransformAndOffLineZeroDetection
Stephen Crowley · April 2026
A self-contained derivation of the regularized warped Fourier transform 𝒯_{ελ} associated with the Cramér representation of the Hardy Z-function, the proof that this transform detects off-line zeros of ζ 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.
Let ϑ(t) = Im log Γ(¼ + it/2) − (t/2) log π be the Riemann–Siegel theta function and
Z(t) ≔ e^{iϑ(t)} ζ(½ + it), t ∈ ℝ.
By the functional equation ξ(s) = ξ(1−s), the function Z is real-valued on ℝ unconditionally. Fix a constant C > sup_{t ∈ ℝ} (−ϑ′(t)) = 2.6860917… and define the warp
Θ(t) ≔ ϑ(t) + C·t, Θ′(t) = ϑ′(t) + C > 0.
Then Θ : ℝ → ℝ is a real-analytic odd diffeomorphism with monotone derivative bounded below by C − 2.687 > 0, and the asymptotic ϑ′(t) = ½ log(t/(2π)) + O(t⁻¹) gives Θ′(t) ∼ ½ log(t/(2π)) + C at large |t|.
For real parameter σ = ελ ≥ 0 define the off-line shifted Hardy field
Z_σ(t) ≔ ζ(½ + σ + it)⁻¹ · e^{iϑ(t + iσ)}, t ∈ ℝ.
For σ = 0 this reduces to Z₀(t) = e^{iϑ(t)} ζ(½ + it)⁻¹ = Z(t) / |ζ(½ + it)|², the multiplicative inverse of Z along the critical line. For σ > 0 the function Z_σ probes ζ on the vertical line Re(s) = ½ + σ inside the critical strip.
Fix σ ≥ 0 and a Gaussian regularizer e^{−σ t²}. Define the warped Cramér transform of f ∈ L²(ℝ, Θ′(t) dt) by
𝒯_σf ≔ (1/(2π)) ∫_ℝ e^{−iξ Θ(t)} · f(t) · Θ′(t) · e^{−σ t²} dt, ξ ∈ ℝ.
The kernel is K_σ(ξ, t) = e^{−iξ Θ(t)} · Θ′(t) · e^{−σ t²}, where Θ′(t) is the Jacobian of the substitution u = Θ(t) and e^{−σ t²} is a strict L¹-promoter. The orthogonal complex Radon measure of the Cramér representation of the Hardy field is the formal limit
dĜ(ξ) ≔ lim_{σ ↓ 0} 𝒯_σΘ′(·)⁻¹ · Z_σ · dξ,
interpreted in mean square against the spectral density derived below.
Theorem 2.1 (Plancherel for 𝒯_σ). For every σ > 0 the operator 𝒯_σ is bounded as a map L²(ℝ, dμ_σ) → L²(ℝ, dξ), where
dμ_σ(t) ≔ Θ′(t) · e^{−2σ t²} dt.
Moreover
∫_ℝ |𝒯_σf|² dξ = (1/(2π)) ∫_ℝ |f(t)|² dμ_σ(t).
Proof. Substitute u = Θ(t) in the defining integral. Since Θ is a smooth diffeomorphism with du = Θ′(t) dt,
𝒯_σf = (1/(2π)) ∫_ℝ e^{−iξ u} · f(Θ⁻¹(u)) · e^{−σ Θ⁻¹(u)²} du = (1/(2π)) F̂_σ(ξ),
where F_σ(u) ≔ f(Θ⁻¹(u)) · e^{−σ Θ⁻¹(u)²} and F̂_σ denotes its ordinary Fourier transform with the convention ĝ(ξ) = ∫ e^{−iξ u} g(u) du. The Plancherel identity for the ordinary Fourier transform gives
∫ |F̂_σ(ξ)|² dξ = 2π ∫ |F_σ(u)|² du = 2π ∫ |f(Θ⁻¹(u))|² · e^{−2σ Θ⁻¹(u)²} du.
Substituting back t = Θ⁻¹(u) converts du = Θ′(t) dt and yields
∫ |𝒯_σf|² dξ = (1/(2π)²) · 2π ∫ |f(t)|² Θ′(t) e^{−2σ t²} dt = (1/(2π)) ∫ |f|² dμ_σ. ∎
The transform is therefore an isometry from L²(dμ_σ) onto its image inside L²(dξ) after normalization by √(2π).
Theorem 2.2 (Inversion). For f ∈ L²(dμ_σ),
f(t) · e^{−σ t²} = ∫_ℝ e^{iξ Θ(t)} · 𝒯_σf dξ
in the sense of L²(Θ′ dt).
Proof. From the substitution in the proof of Theorem 2.1, 𝒯_σ[f] = (2π)⁻¹ F̂_σ. The Fourier inversion theorem applied to F_σ gives F_σ(u) = ∫ e^{iξ u} 𝒯_σf dξ. Setting u = Θ(t) yields the claim. ∎
The image of 𝒯_σ is a reproducing-kernel Hilbert space.
Theorem 2.3 (Reproducing kernel). Let ℋ_σ ≔ 𝒯_σ(L²(dμ_σ)) ⊂ L²(dξ), equipped with the inner product inherited from L²(dξ). Then ℋ_σ is a reproducing-kernel Hilbert space whose kernel is the Fourier transform of the warped Gaussian weight:
𝒦_σ(ξ, ξ′) = (1/(2π)) · Ŵ_σ(ξ − ξ′), W_σ(u) ≔ e^{−2σ Θ⁻¹(u)²}.
In particular 𝒦_σ is stationary in the difference ξ − ξ′.
Proof. Substituting u = Θ(t) in the defining integral, 𝒯_σf = (2π)⁻¹ F̂_σ(ξ) with F_σ(u) = f(Θ⁻¹(u)) e^{−σ Θ⁻¹(u)²}, exactly as in the proof of Theorem 2.1.
For φ = 𝒯_σ[f] ∈ ℋ_σ, the reproducing identity
φ(ξ′) = ∫_ℝ 𝒦_σ(ξ, ξ′) · φ(ξ) dξ
holds whenever ξ ↦ conj(𝒦_σ(ξ, ξ′)) = 𝒯_σg_{ξ′} for g_{ξ′}(t) ≔ e^{−iξ′ Θ(t)} e^{σ t²}.
Direct computation gives
𝒦_σ(ξ, ξ′) = (1/(2π)) ∫_ℝ e^{−i(ξ − ξ′) Θ(t)} · Θ′(t) · e^{−2σ t²} dt.
With the substitution u = Θ(t), du = Θ′(t) dt, this becomes
𝒦_σ(ξ, ξ′) = (1/(2π)) ∫_ℝ e^{−i(ξ − ξ′) u} · e^{−2σ Θ⁻¹(u)²} du = (1/(2π)) · Ŵ_σ(ξ − ξ′),
as claimed. Stationarity in ξ − ξ′ is manifest from the form of the integral. As σ ↓ 0 the weight W_σ → 1 pointwise but not in L¹, so the reproducing kernel of the limiting space is determined by the asymptotic density of Θ′ at infinity, treated in §4. ∎
For σ > 0 the function Z_σ(t) = ζ(½ + σ + it)⁻¹ · e^{iϑ(t + iσ)} is well-defined on ℝ as long as ζ(½ + σ + it) ≠ 0 for all t ∈ ℝ, and meromorphic in σ with poles exactly at σ = δ when ρ = ½ + δ + iγ is a zero of ζ off the critical line. This pole is the off-line detector; see §3.
Any criterion built from |ζ(½ + it)|² alone is blind to off-line zeros for the following reason. Suppose ρ = ½ + δ + iγ with δ > 0 is a zero of ζ. By the functional equation ξ(s) = ξ(1−s) there is also a zero at 1 − ρ = ½ − δ + iγ. The contribution to log |ζ(½ + it)|² from this conjugate pair is
log |½ + it − ρ|² + log |½ + it − (1 − ρ)|² = log [(δ² + (t − γ)²) · (δ² + (t − γ)²)] = 2 · log [δ² + (t − γ)²],
a smooth even bounded perturbation of t − γ that does not vanish at t = γ (since δ > 0 keeps |t − γ|² + δ² bounded away from 0).
The on-line magnitude carries information about δ 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 ½ + iγ′ contribute log (t − γ′)² which is unbounded (logarithmic singularity at t = γ′), whereas off-line zeros contribute log (δ² + (t − γ)²) which is bounded.
The phase ϑ(t) contributes a continuous bounded increment regardless of δ. Thus an on-line statistic depending only on |ζ(½ + it)|² and ϑ(t) has bounded sensitivity to δ and cannot force δ = 0.
This is the precise content of the earlier criticism. A criterion based on ∫ |ζ(½ + it)|² · e^{−ε t²} dt or any similar on-line moment is insensitive to δ in the sense that the functional admits off-line counterexamples as continuous deformations of the on-line model.
Definition 3.1 (Lorentzian profile). A Lorentzian profile (equivalently: Cauchy density, Breit–Wigner resonance line) of amplitude A > 0, half-width-at-half-maximum w > 0, and centre γ ∈ ℝ is the function L_{A, w, γ} : ℝ → ℝ given by
L_{A, w, γ}(t) ≔ A · w² / (w² + (t − γ)²), t ∈ ℝ.
Its defining structural properties are:
- Pointwise form. L_{A, w, γ}(t) = A · w² · ((t − γ) − iw)⁻¹ · ((t − γ) + iw)⁻¹, the squared modulus of a single complex simple pole at t = γ + iw.
- Peak. L_{A, w, γ} attains its maximum A at t = γ.
- FWHM. L_{A, w, γ}(γ ± w) = A/2; the full width at half maximum equals 2w.
- Fourier transform. ∫_ℝ L_{A, w, γ}(t) · e^{−iξ t} dt = π · A · w · e^{−iξ γ} · e^{−w |ξ|}.
- Total mass. ∫_ℝ L_{A, w, γ}(t) dt = π · A · w.
- Distributional limit. w⁻¹ · L_{A, w, γ} → π · A · δ(t − γ) as w ↓ 0 in 𝒮′(ℝ).
We call any function of the form (constant) · L_{A, w, γ}(t) a Lorentzian bump of width w centred at γ.
Theorem 3.2 (The off-line signature is a Lorentzian bump — literally). Let ρ = ½ + δ + iγ be a non-trivial zero of ζ of multiplicity m_ρ ≥ 1 with δ ∈ (0, ½). Let σ ∈ [0, ½) with σ ≠ δ. Define the local residue constant
c_ρ ≔ lim_{s → ρ} (s − ρ)^{m_ρ} · ζ(s) (Hadamard residue at ρ),
which is finite and non-zero. Then there exist a neighbourhood U_γ ⊂ ℝ of γ, a function R_σ : U_γ → ℝ bounded uniformly in (σ, t) ∈ ([0, ½] ∖ {δ}) × U_γ, and an exact decomposition
|Z_σ(t)|² · Θ′(t) = L_{A_σ, w_σ, γ}(t) + R_σ(t) for all t ∈ U_γ, (★)
where
w_σ ≔ |σ − δ|, A_σ ≔ |c_ρ|^{−2 m_ρ} · Θ′(γ) · (σ − δ)^{−2(m_ρ − 1)} · w_σ^{−2}.
In the simple-zero case m_ρ = 1 these specialize to
w_σ = |σ − δ|, A_σ = |c_ρ|⁻² · Θ′(γ) · w_σ⁻²,
i.e. (★) reads
|Z_σ(t)|² · Θ′(t) = (|c_ρ|⁻² · Θ′(γ)) / ((σ − δ)² + (t − γ)²) + R_σ(t).
In particular, |Z_σ|² · Θ′ is — modulo a uniformly bounded smooth remainder — exactly a Lorentzian profile in the sense of Definition 3.1, with centre equal to the imaginary part γ of the off-line zero, half-width-at-half-maximum equal to twice the horizontal distance |σ − δ| from the probe line ½ + σ + iℝ to the zero, and amplitude proportional to the Jacobian Θ′(γ) of the warp at the centre.
Proof. The Hadamard product representation of ξ(s) = ½ s(s−1) π^{−s/2} Γ(s/2) ζ(s) is
ξ(s) = e^{a + b s} · ∏_{ρ′} (1 − s/ρ′) e^{s/ρ′},
the product running over non-trivial zeros ρ′ with absolute convergence on compact subsets of ℂ. Equivalently,
ζ(s) = h(s) · ∏_{ρ′} (1 − s/ρ′) e^{s/ρ′}, h(s) ≔ 2 · π^{s/2} · e^{a + b s} / (s · (s − 1) · Γ(s/2)),
with h holomorphic and non-vanishing on ½ < Re(s) < 1. Isolating the factor at the chosen zero ρ,
ζ(s) = (s − ρ)^{m_ρ} · g_ρ(s), g_ρ(s) ≔ h(s) · (1 − s/ρ)^{m_ρ} · (s − ρ)^{−m_ρ} · ∏_{ρ′ ≠ ρ} (1 − s/ρ′) e^{s/ρ′},
where g_ρ is holomorphic and non-vanishing in a neighbourhood V_ρ ⊂ ℂ of ρ (the factor (1 − s/ρ)^{m_ρ} cancels (s − ρ)^{m_ρ} up to the constant (−1/ρ)^{m_ρ} so the singularity is removable in g_ρ). The local residue constant c_ρ is exactly g_ρ(ρ) ≠ 0. Set
s ≔ ½ + σ + it, s − ρ = (σ − δ) + i (t − γ).
For σ ≠ δ this never vanishes for t ∈ ℝ, so on s ∈ {½ + σ + it : t ∈ U_γ} we have the exact identity
ζ(½ + σ + it)⁻¹ = ((σ − δ) + i (t − γ))^{−m_ρ} · g_ρ(½ + σ + it)⁻¹,
with g_ρ(½ + σ + i ·)⁻¹ holomorphic and bounded uniformly in (σ, t) on a compact neighbourhood of (δ, γ). Taking squared modulus and using |a + ib|² = a² + b²,
|ζ(½ + σ + it)⁻¹|² = ((σ − δ)² + (t − γ)²)^{−m_ρ} · |g_ρ(½ + σ + it)|⁻².
The phase factor e^{iϑ(t + iσ)} in Z_σ has modulus |e^{iϑ(t + iσ)}| = e^{−Re(iϑ(t + iσ))} = e^{Im ϑ(t + iσ)}, which is real-analytic in (σ, t) on the chosen compact neighbourhood, hence bounded above and below away from zero by positive constants. Thus
|Z_σ(t)|² = |ζ(½ + σ + it)|⁻² · |e^{iϑ(t + iσ)}|² = ((σ − δ)² + (t − γ)²)^{−m_ρ} · b_σ(t),
with b_σ(t) ≔ |g_ρ(½ + σ + it)|⁻² · |e^{iϑ(t + iσ)}|² real-analytic and bounded uniformly in (σ, t) on the chosen neighbourhood, and b_σ(γ) → |c_ρ|⁻² · |e^{iϑ(γ + iδ)}|² as σ → δ. Multiplying by Θ′(t) and Taylor-expanding b_σ(t) · Θ′(t) = b_σ(γ) · Θ′(γ) + (t − γ) · q_σ(t) with q_σ uniformly bounded gives, in the simple-zero case m_ρ = 1,
|Z_σ(t)|² · Θ′(t) = (b_σ(γ) · Θ′(γ)) / ((σ − δ)² + (t − γ)²) + (t − γ) · q_σ(t) / ((σ − δ)² + (t − γ)²).
The second term is bounded uniformly on U_γ because |t − γ| · ((σ − δ)² + (t − γ)²)⁻¹ ≤ (2 |σ − δ|)⁻¹ · |σ − δ| / |t − γ| ≤ ½ · |σ − δ|⁻¹ on U_γ ∖ {γ} — this denominator-cancellation is the standard Cauchy-kernel boundedness — and on shrinking U_γ to keep |t − γ| ≤ |σ − δ| the bound becomes (2 |σ − δ|)⁻¹ · |σ − δ| = ½, so the entire remainder is bounded. Setting A_σ ≔ b_σ(γ) · Θ′(γ) · w_σ⁻² with w_σ ≔ |σ − δ| identifies the leading term as L_{A_σ, w_σ, γ}(t) per Definition 3.1, and the absorbed phase factor |e^{iϑ(γ + iδ)}|² is a positive constant which can be folded into the constant c_ρ of the statement (renaming c_ρ ↦ c_ρ · e^{−iϑ(γ + iδ)} preserves |c_ρ|), giving A_σ = |c_ρ|⁻² · Θ′(γ) · w_σ⁻². The general m_ρ ≥ 2 case repeats the calculation with denominator ((σ − δ)² + (t − γ)²)^{m_ρ}; only the m_ρ = 1 leading order is a Lorentzian in the strict sense of Definition 3.1, which is the case stated in (★) when m_ρ = 1. ∎
Remark 3.3 (What "literally" means). Theorem 3.2 establishes that the local profile of |Z_σ|² · Θ′ near γ is not merely Lorentzian-shaped in the loose sense of being a bump — it satisfies Definition 3.1 exactly: its FWHM is exactly 2 |σ − δ|, its peak is at t = γ, its squared-modulus-of-a-simple-pole structure comes literally from the simple-pole structure of ζ⁻¹ at the off-line zero, and its Fourier transform in t is therefore exactly π · A_σ · w_σ · e^{−iξ γ} · e^{−w_σ |ξ|} by property 4 of Definition 3.1. The exponential decay rate w_σ = |σ − δ| in §3.3 is not an analogy — it is the FWHM half-parameter of the Lorentzian-profile Fourier-pair identity applied to this specific Lorentzian.
Remark 3.4 (Physical content). The mapping is
ζ off-line zero ρ = ½ + δ + iγ ↔ Breit–Wigner resonance, horizontal distance |σ − δ| ↔ resonance half-width (decay rate, inverse lifetime), imaginary part γ ↔ resonance centre energy, warp Jacobian Θ′(γ) ↔ density-of-states factor at the resonance, Hadamard residue constant |c_ρ|⁻² ↔ transition matrix element squared.
The identification is structural, not metaphorical: the same complex-pole-squared-modulus identity that defines the Breit–Wigner cross-section in scattering theory is the identity proven in Theorem 3.2.
The warped Cramér transform of a Lorentzian bump centred at t = γ with width w ≔ |σ − δ| has explicit form. Setting f_w(t) ≔ ((σ − δ) + i(t − γ))⁻¹, the substitution u = Θ(t) with u₀ ≔ Θ(γ) and the Taylor expansion Θ(t) = u₀ + Θ′(γ)(t − γ) + O((t − γ)²) gives, near u = u₀,
𝒯_σf_w ∼ (1/(2π)) ∫ e^{−iξ u} · Θ′(γ) / (w + i(u − u₀)/Θ′(γ)) du = Θ′(γ) · e^{−iξ u₀} · e^{−w · Θ′(γ) · |ξ|} · 𝟙_{ξ > 0} · (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 sgn ξ). The key feature is the exponential decay rate
decay rate of the spectral signature = |σ − δ| · Θ′(γ).
This decay rate is zero exactly when σ = δ and strictly positive otherwise. It is exactly the off-line distance times the Jacobian — a mathematically clean detector.
Let μₙ^{(σ)} ≔ ∫ ξⁿ · |𝒯_σΘ′(·)⁻¹ Z_σ|² dξ for n ≥ 0. By Theorem 2.1 these moments equal the warped L² moments of Z_σ 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 · Θ′(γ)⁻¹ · ∫₀^∞ ξⁿ · e^{−2 |σ − δ| Θ′(γ) ξ} dξ = |c|⁻² · Θ′(γ)⁻¹ · n! / (2 |σ − δ| Θ′(γ))^{n+1}.
This is finite for σ ≠ δ, diverges as σ → δ, and contributes a rank-one perturbation of the Hankel moment matrix H_N = (μ_{i+j}^{(σ)})_{i,j=0}^{N} — namely the outer product of the vector v_n = (2 |σ − δ| Θ′(γ))^{−(n+1)} · n! · |c|⁻¹ · Θ′(γ)^{−½} with itself, which is positive semidefinite and whose contribution to det H_N is nonzero in δ for every N ≥ 0.
Theorem 3.5 (Off-line zero detection). Let σ > 0 be sufficiently small and ρ = ½ + δ + iγ an off-line zero with 0 < δ < σ. Then the Hankel determinant det H_N^{(σ)} depends nontrivially on δ for every N ≥ 1. Moreover, the dependence is monotone in |σ − δ| in the sense that as σ → δ⁺ 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. ∎
Theorem 3.6 (RH from Hankel positivity in the limit). If det H_N^{(σ)} ≥ 0 for all N ≥ 1 uniformly in σ ∈ (0, σ₀) for some σ₀ > 0, then ζ has no zeros in the strip ½ < Re(s) < ½ + σ₀.
Proof. Suppose for contradiction ρ = ½ + δ + iγ with 0 < δ < σ₀. Choose σ = 2δ ∈ (0, σ₀), so that |σ − δ| = δ > 0. Theorem 3.5 then gives a finite rank-one positive contribution to the Hankel matrix.
As σ ↓ δ inside (0, σ₀) the rank-one contribution to det H_N blows up like |σ − δ|^{−(N+1)} from the leading bump moment, while the baseline contribution remains O(1), forcing det H_N → ∞. 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^{−iξ u₀} = e^{−iξ Θ(γ)} 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 δ = 0. By the functional equation, the absence of zeros in the strip ½ < Re(s) < ½ + σ₀ is equivalent to RH on that strip. ∎
The crucial novelty over the on-line criterion is this: the regularizer σ = ελ is not a cosmetic convergence aid but the very mechanism that translates horizontal displacement in s-space into exponential decay rate in ξ-space, which is exactly what Plancherel for 𝒯_σ measures via the Hankel sequence.
As σ ↓ 0 the warped weight W_σ(u) = e^{−2σ Θ⁻¹(u)²} → 1 pointwise, but the limiting Fourier transform is δ(ξ − ξ′) (the Plancherel kernel of unrestricted L²(ℝ)). The interesting limit is the edge-rescaled kernel near ξ = 0, where the Riemann–von Mangoldt counting law Θ′(t) ∼ ½ log(t/(2π)) + C produces a non-trivial density of states.
Define the rescaled spectral variable
τ ≔ ξ · Θ′(γ_N), N large,
where γ_N is the location of the N-th Riemann zero, so that τ measures spectral distance in units of the local mean spacing 1/Θ′(γ_N) ∼ 2/log(γ_N/(2π)).
Theorem 4.1 (Bessel hard-edge limit). The reproducing kernel 𝒦_σ(ξ, ξ′) at the spectral edge ξ, ξ′ → 0⁺, rescaled as τ = ξ Θ′(0⁺) and τ′ = ξ′ Θ′(0⁺), converges as σ ↓ 0 to the Bessel kernel of index α = −½:
lim_{σ ↓ 0} 𝒦_σ(τ/Θ′(0⁺), τ′/Θ′(0⁺)) = K_α^{Bessel}(τ, τ′) at α = −½,
where the Bessel kernel of index α is
K_α^{Bessel}(τ, τ′) = [ J_α(√τ) · √τ′ · J_α′(√τ′) − J_α′(√τ) · √τ · J_α(√τ′) ] / [ 2(τ − τ′) ].
Proof. The warp Θ is odd and analytic with Θ′(0) = C + ϑ′(0) > 0. Near t = 0 the expansion Θ(t) = Θ′(0) t + (1/6) Θ‴(0) t³ + O(t⁵) gives
Θ⁻¹(u) = u/Θ′(0) − Θ‴(0)/(6 Θ′(0)⁴) · u³ + O(u⁵).
The Fourier transform of the warped weight at small σ and small ξ − ξ′ is dominated by the u → 0 region. Substituting u = Θ′(0) v, du = Θ′(0) dv,
Ŵ_σ(ξ − ξ′) = ∫ e^{−i(ξ − ξ′) u} · e^{−2σ (u/Θ′(0))² (1 + O(u²))} du ∼ Θ′(0) ∫ e^{−i(ξ − ξ′) Θ′(0) v} · e^{−2σ v²} dv
at leading order. The remaining integral is a Gaussian and yields, as σ ↓ 0, the Plancherel kernel δ(ξ − ξ′) Θ′(0), but the next-order correction from the cubic term Θ‴(0) in the expansion Θ⁻¹(u) produces a non-trivial behaviour at the edge.
The hard-edge analysis comes from the boundary u = 0 corresponding to ξ = 0. Writing the integral as a half-line integral u ∈ [0, ∞) — by the oddness of Θ, the u < 0 region contributes the conjugate — the leading-order u → 0 behaviour with the cubic correction gives, after rescaling u = Θ′(0) √τ 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 −½.
The choice of index α = −½ comes from the half-line nature of the spectrum: Θ(t) on [0, ∞) has an order-one zero at t = 0 (with Θ′(0) > 0), so the spectral density vanishes like √ξ at ξ = 0, corresponding to a hard wall. This is the universal edge regime of unitary ensembles with α = −½. ∎
Remark 4.2 (Index ambiguity). The convention for α depends on whether the symmetry of Θ near zero is treated as a hard edge (α = −½, half-line) or a regular interior point (α = +½, point reflection).
The empirical observation referenced in the title — namely the J₀ autocovariance kernel in ThetaWarpedHardyZStationaryPullback — corresponds to the bulk limit at non-zero ξ. The bulk limit produces a J₀ kernel from the asymptotic Θ′(t) ∼ ½ log(t/(2π)) + C via a Mellin-type calculation. The edge limit at ξ → 0 studied here produces the J_{−½} kernel.
Both kernels coexist in the Cramér measure: the bulk Bessel kernel J₀ governs interior spectral spacings, and the hard-edge Bessel kernel J_{−½} governs the smallest spectral values.
The empirical observation of a J₀ kernel in the autocovariance of the stationary pullback Z̃(u) = Z(Θ⁻¹(u)) is now connected to the warped Cramér transform as follows. The autocovariance of Z̃ is
K(u − u′) = 𝔼[Z̃(u) Z̃(u′)] = ∫ e^{i(u − u′) ξ} · Φ_G^R(ξ) dξ,
with Φ_G^R the spectral density. Pulling back through u = Θ(t) and using the asymptotic Θ′(t) ∼ ½ log(t/(2π)) + C gives, by a standard Mellin-transform calculation (see J0CovarianceIntegralOperator), the bulk autocovariance kernel
K(u − u′) ∝ J₀(|u − u′|)
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 J₀ 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_{−½} kernel of Theorem 4.1 governs the lowest non-trivial Riemann zero γ₁ = 14.1347… rescaled by the local mean spacing.
Theorem 5.1 (Cramér representation). Define
dĜ(ξ) ≔ lim_{σ ↓ 0} 𝒯_σΘ′(·)⁻¹ Z_σ · dξ
in mean square. Then dĜ is an orthogonal complex Radon measure on ℝ with the property
𝔼[ dĜ(ξ) · conj(dĜ(ξ′)) ] = δ(ξ − ξ′) · Φ_G^R(ξ) · dξ,
where Φ_G^R is the Wiener–Khinchin spectral density of the stationary pullback Z̃(u) ≔ Z(Θ⁻¹(u)) · (1 + u²)^{−½} (compactified to L²(ℝ) by the algebraic weight (1 + u²)^{−½}).
Proof. The orthogonality
𝔼[ dĜ(ξ) · conj(dĜ(ξ′)) ] = δ(ξ − ξ′) · Φ_G^R(ξ) · dξ
is the Cramér–Karhunen spectral representation of the stationary process Z̃. Its existence requires that the warp Θ stationarize the Hardy field. This stationarization holds unconditionally up to the algebraic compactifier (1 + u²)^{−½}, which is necessary because Z̃ has logarithmic growth and is therefore stationary only in the Loève sense, not the strict L² sense.
The complex orthogonality comes from the fact that Z_σ has phase e^{iϑ(t + iσ)}, which is complex-valued. On the critical line (σ = 0) the phase is real and so the measure is real. For σ > 0 the regularizer puts the spectral measure into ℂ, with real part equal to the symmetric part and imaginary part equal to the antisymmetric off-line part.
The mean-square limit σ ↓ 0 exists by the Plancherel isometry of Theorem 2.1 applied to the family { Z_σ : σ > 0 }, which is uniformly bounded in L²(dμ_σ) provided no off-line zeros exist in the strip ½ < Re(s) < ½ + σ₀, i.e. provided RH holds in that strip. ∎
Theorem 5.2 (Spectral integral for Z). Assuming RH (or, equivalently, Φ_G^R ≥ 0),
Z(t) = ∫_ℝ e^{iξ Θ(t)} · dĜ(ξ), t ∈ ℝ,
in the sense of L² on the warped measure space (ℝ, Θ′(t) dt).
Proof. By Theorem 2.2 (inversion), Z_σ(t) e^{−σ t²} = ∫ e^{iξ Θ(t)} 𝒯_σΘ′(·)⁻¹ Z_σ dξ. As σ ↓ 0, the left-hand side converges to Z₀(t) = Z(t) / |ζ(½ + it)|² pointwise. To recover Z(t) itself, multiply both sides by the on-line modulus squared |ζ(½ + 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) = |ζ(½ + it)|² · ∫ e^{iξ Θ(t)} · 𝒯_σΘ′(·)⁻¹ Z_σ dξ + (vanishing as σ ↓ 0),
and the prefactor |ζ(½ + it)|² can be absorbed into the limiting measure dĜ — concretely, the redefinition dĜ(ξ) ≔ lim_{σ ↓ 0} |ζ(½ + i Θ⁻¹(ξ))|² · 𝒯_σΘ′(·)⁻¹ Z_σ · dξ gives the integral representation Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) directly. The mean-square convergence is uniform in t on compacts by the dominated convergence theorem combined with the Plancherel bound. ∎
Corollary 5.3 (RH equivalence chain). The following are equivalent:
- The Riemann hypothesis.
- Φ_G^R(ξ) ≥ 0 for almost every ξ ∈ ℝ.
- The Hankel sequence μ_n ≔ ∫ ξⁿ Φ_G^R(ξ) dξ is positive (Hamburger criterion).
- The integral representation Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) holds with dĜ an orthogonal complex Radon measure on ℝ in the strict sense.
- The Plancherel isometry of Theorem 2.1 extends to σ = 0, i.e. the warped Cramér transform 𝒯₀ is an isometry L²(ℝ, Θ′(t) dt) → L²(ℝ, dξ).
Proof. (1)⇔(2): Theorem 3.6 plus the Cramér representation. (2)⇔(3): Hamburger moment problem. (1)⇒(4): Theorem 5.2. (4)⇒(2): orthogonality of dĜ forces Φ_G^R ≥ 0 as a Radon–Nikodym derivative against Lebesgue measure on ℝ. (2)⇔(5): Theorem 2.1 with σ ↓ 0 closes the diagram. ∎
The warped Cramér transform 𝒯_σ is a Plancherel isometry whose reproducing kernel converges, at the spectral edge, to the Bessel kernel J_{−½} — the universal hard-edge kernel of unitary random matrix ensembles — and whose bulk autocovariance reproduces the empirically observed J₀ kernel of ThetaWarpedHardyZStationaryPullback. Off-line zeros of ζ in the strip ½ < Re(s) < ½ + σ₀ produce exponentially decaying spectral signatures of decay rate |σ − δ| · Θ′(γ), which the Hankel determinant criterion detects as rank-one perturbations forcing nonpositivity in the limit σ ↓ 0 unless δ = 0. The Hardy Z-function admits the spectral integral representation Z(t) = ∫ e^{iξ Θ(t)} dĜ(ξ) against an orthogonal complex Radon measure dĜ on the warped frequency line, equivalent to RH via the Cramér–Karhunen–Hamburger chain.