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The Riemann Hypothesis as a Stability Principle

  The Riemann Hypothesis can be reformulated as an infinite sequence of explicit positivity tests. This is not a proof. It turns a statement about complex zeros into inequalities that can be calculated, falsified at finite order, and compared with operator or physical models.   The completed zeta function is $$\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s),$$ with $$\Xi(t)=\xi\left(\frac12+it\right).$$ The function $\Xi$ is even and real on the real axis. A nontrivial zero $\rho=\beta+i\gamma$ corresponds to $$t_\rho=\gamma-i\left(\beta-\frac12\right).$$ Therefore $$t_\rho\in\mathbb R \quad\Longleftrightarrow\quad \beta=\frac12,$$ and hence $$\boxed{\mathrm{RH}\quad\Longleftrightarrow\quad\text{every zero of }\Xi\text{ is real}.}$$ Folding the Zeros Because $\Xi$ is even, write $$\Xi(t)=\sum_{n\ge0}c_nt^{2n}.$$ Now define $$G(z)=\frac{\Xi(i\sqrt z)}{\Xi(0)}.$$ Although this contains $\sqrt z$, the even expansion makes $G$ entire: $$G(z)=...
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Riemann Zeros from the Edge of Spacetime

This not a proof of the Riemann Hypothesis (RH). Its purpose is to explore a simple idea: that horizons supply a natural quantum clock.

Black-hole horizons and the horizon of de Sitter space share a simple property: they stretch a light-ray coordinate exponentially. That stretching supplies a natural quantum clock. If the clock is prepared in one special state, its return amplitude is the Riemann $\Xi$-function. The real zeros of $\Xi$ then mark exact times when the state has become orthogonal to where it began.

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This idea does not turn the zeros into energy levels. It gives them a different physical meaning: they are horizon times. 

A Horizon That Stretches Light

Let $V>0$ be an affine coordinate along a horizon light ray, and let $\kappa>0$ be the surface gravity. Horizon time acts as

$$V(t)=e^{\kappa t}V.$$

A wavefunction $\psi(V)\in L^2(\mathbb R_+ dV)$ has norm $\int_0^\infty |\psi(V)|^2 dV$. Probability conservation requires

$$(U_t\psi)(V) = e^{-\kappa t/2}\psi(e^{-\kappa t}V).$$

Writing $U_t=e^{-i\widehat Ht/\hbar}$ gives

$$\widehat H = -i\hbar\kappa \left( V\frac{d}{dV}+\frac12 \right).$$

The factor $1/2$ is the Jacobian term required by unitarity.

Introduce logarithmic horizon position,

$$q=\ln(V/V_0), \qquad \chi(q)=\sqrt V \psi(V).$$

Since $dV=V dq$,

$$\int_0^\infty |\psi(V)|^2 dV = \int_{-\infty}^{\infty}|\chi(q)|^2 dq $$

In this coordinate,

$$\widehat H = -i\hbar\kappa\frac{d}{dq}$$

and $\chi(q,t)=\chi(q-\kappa t,0)$. Exponential stretching in $V$ has become ordinary translation in $q$.

The generalized eigenmodes are

$$\langle V|\lambda\rangle = \frac{1}{\sqrt{2\pi}} V^{-1/2} \exp\left( i\lambda\ln\frac{V}{V_0} \right),$$

with energies $E_\lambda=\hbar\kappa\lambda$. The horizon spectrum is continuous.

Preparing the Riemann State

Define

$$\xi(s) = \frac12s(s-1)\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s),$$

and $\Xi(u) = \xi\left(\frac12+iu\right)$. For real $u$,

$$\Xi(u) = \int_{-\infty}^{\infty} \Phi(\lambda)e^{-iu\lambda} d\lambda,$$

where $\Phi$ is real, even, positive and rapidly decreasing. Therefore

$$p_\Xi(\lambda) = \frac{\Phi(\lambda)}{\Xi(0)}$$

is a normalised probability density: $\int_{-\infty}^{\infty}p_\Xi(\lambda) d\lambda=1$.

Prepare the state

$$|\Omega_\Xi\rangle = \int_{-\infty}^{\infty} \sqrt{p_\Xi(\lambda)} |\lambda\rangle d\lambda.$$

The square root is essential: $p_\Xi(\lambda)$ is the probability density, while $\sqrt{p_\Xi(\lambda)}$ is the quantum amplitude.

The Zeros Become Times

Each mode evolves as $|\lambda\rangle \longmapsto e^{-i\kappa t\lambda}|\lambda\rangle$. Therefore

$$\mathcal A(t) = \langle\Omega_\Xi| e^{-i\widehat Ht/\hbar} |\Omega_\Xi\rangle = \int_{-\infty}^{\infty} p_\Xi(\lambda)e^{-i\kappa t\lambda} d\lambda = \frac{\Xi(\kappa t)}{\Xi(0)}.$$

If $\gamma_n$ is a real zero of $\Xi$, then $\mathcal A\left(\gamma_n/\kappa\right)=0$. Thus the state becomes orthogonal to its initial state at

$$\boxed{\kappa t_n=\gamma_n.}$$

In logarithmic position space, define

$$\chi_\Xi(q) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \sqrt{p_\Xi(\lambda)} e^{i\lambda q} d\lambda.$$

Then

$$\mathcal A(t) = \int_{-\infty}^{\infty} \chi_\Xi^*(q) \chi_\Xi(q-\kappa t) dq$$

The zeros are displacements at which the logarithmic horizon wavepacket becomes orthogonal to its translated copy.

Complex Horizon Time and the Riemann Hypothesis

Because $\Phi$ decreases rapidly, the amplitude has an entire continuation:

$$\mathcal A_{\mathbb C}(t) = \frac{ \xi\left(\frac12+i\kappa t\right) }{ \xi\left(\frac12\right) }, \qquad t\in\mathbb C.$$

Let $\rho=\beta+i\gamma$ be a nontrivial zero of $\zeta$. The corresponding amplitude zero satisfies $\frac12+i\kappa t_\rho=\rho$. Hence

$$\boxed{ \kappa t_\rho = \gamma-i\left(\beta-\frac12\right). }$$

Therefore $\operatorname{Re}(\kappa t_\rho)=\gamma$ and $\operatorname{Im}(\kappa t_\rho)=\frac12-\beta$. It follows that

$$\operatorname{Im}t_\rho=0 \quad\Longleftrightarrow\quad \beta=\frac12.$$

Thus

$$\boxed{ \mathrm{RH} \quad\Longleftrightarrow\quad \text{every zero of }\mathcal A_{\mathbb C}(t) \text{ occurs at real Lorentzian time}. }$$

A critical-line zero is a physical orthogonality event. An off-line zero would occur only at complex time, where the amplitude is an analytic continuation rather than a unitary overlap.

Since nontrivial zeros satisfy $0<\beta<1$, the critical strip becomes

$$\left|\operatorname{Im}t\right| < \frac{1}{2\kappa}.$$

The Riemann Hypothesis says that every amplitude zero in this strip lies on its real-time centre.

The de Sitter Horizon

Nothing above requires a black-hole singularity. It uses only stationary horizon dilation.

For de Sitter space,

$$ds^2 = -\left(1-H_{\mathrm{dS}}^2r^2\right)dt^2 + \frac{dr^2}{1-H_{\mathrm{dS}}^2r^2} + r^2d\Omega^2,$$

the cosmological horizon lies at $r_c=H_{\mathrm{dS}}^{-1}$, with surface gravity $\kappa=H_{\mathrm{dS}}$. Therefore

$$\mathcal A_{\mathrm{dS}}(t) = \frac{\Xi(H_{\mathrm{dS}}t)}{\Xi(0)},$$

and $t_n^{\mathrm{dS}} = \gamma_n/H_{\mathrm{dS}}$. The universal variable is dimensionless horizon time, $\tau=\kappa t$. For a thermal horizon, $k_BT_H=\hbar\kappa/(2\pi)$, so

$$\frac{k_BT_Ht_\rho}{\hbar} = \frac{\gamma}{2\pi} - \frac{i}{2\pi} \left(\beta-\frac12\right).$$

What is new with this idea?

Spectral proposals impose boundary conditions that discretise a dilation Hamiltonian and interpret the Riemann zeros as energy levels. Here the dilation spectrum remains continuous. The arithmetic information is encoded instead in one state's spectral density. This differs from the boundary-condition construction in Black Holes, Quantum Chaos, and the Riemann Hypothesis.

Other quantum constructions produce zeta-related Loschmidt amplitudes through engineered states, spectra and interactions. Here the horizon supplies the translation generator directly.

Thus

$$\boxed{ \text{critical-line Riemann zero} \quad\longleftrightarrow\quad \text{horizon orthogonality time}. }$$

More fully,

$$\boxed{ \mathrm{RH} \quad\Longleftrightarrow\quad \text{all zeros of the continued horizon coherence lie on the unitary-time axis}. }$$

What This Does Not Explain

The horizon supplies the dilation dynamics, but the special state supplies the arithmetic content. Our framework does not explain why gravity would prepare

$$p_\Xi(\lambda) = \frac{\Phi(\lambda)}{\Xi(0)}.$$

It simply proves what follows if that state is prepared. It does not prove the Riemann Hypothesis, and complex time is not directly observable.

 

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