Why the Riemann Hypothesis and Fluid Turbulence May Be the Same Problem

Tesla coils, Majorana towers, and the spectral fingerprint connecting two Millennium Prize problems.

In my free time, I used to build audio-modulated solid state Tesla coils as a hobbyist. The physics of the plasma streamers that came off those coils fascinated me, and as it turns out, that physics is closely related to the physics of fluid turbulence. Plasmas obey a similar mathematical description. What is interesting is that two of the Clay Institute's Millennium Prize Problems, the Riemann hypothesis on one hand and the problem of intermittency in fluid turbulence on the other, appear to be related to each other.

Presentation

https://www.youtube.com/watch?v=U9CuORupmS8

How can it be that two of the largest open problems in physics and mathematics are saying the same thing? That is what I want to lay out here.

The Riemann Hypothesis as Physics

The Riemann zeta function encodes the distribution of prime numbers. Prime factorization is, of course, the bedrock of current cryptographic standards, which rely on the difficulty of factoring large primes. That difficulty can be sidestepped by quantum computers running Shor's algorithm, or, with quadratic speedup, Grover's algorithm. The Riemann hypothesis asks whether all of the non-trivial zeros of this function lie along the so-called critical line, with real part 1/2. Numerical search has verified this for the first many billions of zeros. None has been found off the line. But the conjecture remains a conjecture.

What physicists noticed, and what is the doorway into everything that follows, is that this function looks suspiciously like it could be describing the energy levels of some quantum mechanical system. This is the Hilbert-Pólya conjecture: the idea that there exists a self-adjoint operator whose eigenvalues coincide with the imaginary parts of the non-trivial zeta zeros. Because self-adjoint operators have real eigenvalues, the existence of such an operator would automatically prove the Riemann hypothesis. The hypothesis would then be settled not by abstract mathematics but by physics.

This is not idle speculation. In 1973, Hugh Montgomery showed that the pair correlations of the non-trivial zeta zeros match the Gaussian Unitary Ensemble statistics of random matrix theory, the very same statistics that govern eigenvalue spacing in quantum chaotic Hamiltonians with broken time-reversal symmetry. Andrew Odlyzko verified this numerically to extraordinary precision for the first billion zeros. In 1999, Berry and Keating elaborated the picture: the underlying "Riemann dynamics" should be chaotic, with periodic orbits labeled by primes. They proposed the Berry-Keating Hamiltonian, H=(xp+px)/2H = (xp + px)/2 H=(xp+px)/2, as a candidate for the Hilbert-Pólya operator.

Then, in 2021, Tamburini and Licata showed something remarkable. The Majorana equation placed in Rindler spacetime, the frame of a uniformly accelerated observer, reduces to a Hamiltonian of exactly the Berry-Keating form. The boundary conditions on the Majorana wavefunction involve Bessel functions which, through Mellin-Barnes integrals, connect directly to the Riemann zeta function. Their 2025 preprint extends this to a claimed conditional proof of the Hilbert-Pólya conjecture: the Majorana Hamiltonian in Rindler space is essentially self-adjoint, with eigenvalues in bijective correspondence with the imaginary parts of non-trivial zeta zeros. I have had many long conversations with Dr. Tamburini about this, and I think it deserves much more attention than it has received.

Why Fluid Turbulence Wears the Same Statistics

Here is where the physics widens. The Riemann zeta function does not just show up in random matrix theory and accelerated quantum field theory. It shows up in the statistics of nonlinear deterministic systems near phase transitions. Quantum chaos, fluid turbulence intermittency, magnetohydrodynamic instabilities. All of them.

Modeling fluid turbulence and atmospheric flows, or the plasma behavior in fusion tokamaks and coronal mass ejections, is largely intractable. The physics is not well understood. So if there really is a universal spectral signature governing these systems, and it really is the Riemann zeta function, then resolving this would do considerably more than collect two million dollars from the Clay Institute. It would help us understand magnetohydrodynamic instabilities, which are the bottleneck for plasma confinement in tokamaks. It would improve weather prediction. And, since the same physics sits at the intersection of nonlinear deterministic and quantum probabilistic regimes, it could illuminate quantum gravity itself. Quantum gravity is, in a sense, the missing physics of reconciling those two regimes. As I have discussed with Ed Witten, macroscopic quantum-like behavior is one direction for understanding it.

The Majorana Tower and the Cascade

So what is going on physically? One clue comes from Majorana fermions and Majorana zero modes. These are particles that are their own antiparticles, and they enjoy what is called topological protection, which makes them the leading candidate for noise-resilient scalable quantum computation. Microsoft is actively investing in this technology for that reason. There are also theories which link Majorana physics to how the brain processes information, something I have discussed at length with Dr. Tamburini.

The picture I find compelling is this. Fermionic spin systems can be entangled and pumped by Floquet drivers into a state of saturation. Beyond that point, the system undergoes a phase transition at which its statistics are described by the Riemann zeta function. The information stored in the entanglement structure is then bosonized into ultraweak superradiant Majorana-like vortex photon signals carrying orbital angular momentum (OAM). There may also be gravitational feedback at these phase transitions.

This sounds speculative. It is, at the edges. But the load-bearing pieces are not. There are now multiple publications reporting that the zeros of the Riemann zeta function have been reproduced experimentally by pumping or driving qubits, and the mathematical link to Majorana physics has appeared in respectable journals. OAM light is known to carry the kind of quantum information needed for these effects, and has been explored as a candidate explanation for the black hole information paradox and for intracellular communication in living tissue.

One way to read all this: the boundary between the right and left Rindler wedges, in collections of entangled Majorana particles, traces out the critical line of the Riemann zeta function. The split structure of these particles is how quantum information gets stored before being saturated and released as light and gravity. The mathematics here implicates Z2 orbifolds at the phase transition, the bosonization of information into light-like modes, and the resulting superradiant cascades. Twistor theory, the theory of null geodesics that Penrose has long argued must be central to a complete theory of gravity, fits naturally into this picture. So does Einstein-Cartan theory, which introduces spacetime torsion to general relativity, or, equivalently, gravity to quantum theory through spin. Could these light-like signals be implicated in the transmission of the gravitational force? Beyond-standard-model physics has been investigated along exactly these lines as a direction for understanding dark matter and dark energy, through the seesaw mechanism and the unique state oscillations these particles allow.

The Original Majorana Tower

Most physicists know Majorana for the Majorana spinor, the equation, and the zero modes. Fewer know that Majorana's 1932 paper introduced an infinite-component relativistic wave equation, the first construction of infinite-dimensional unitary representations of the Lorentz group, predating Wigner and Bargmann by years. The equation yields an infinite tower of particle states with increasing spin and a characteristic mass spectrum. Unlike the Dirac equation, all energies are positive. Majorana's original motivation, in fact, was eliminating the negative-energy sea. Higher-spin states have lower effective mass, creating a natural hierarchy of states across scales.

Tamburini and collaborators showed that this extends to photons with orbital angular momentum propagating in structured plasmas via the Anderson-Higgs mechanism. The key point for turbulence is this: the Majorana tower describes an infinite set of states where higher angular momentum corresponds to lower effective mass, a natural hierarchy mirroring the multi-scale cascade structure of turbulence, where energy is transferred from large eddies to small. That is to say, eddies of low angular momentum at large scale, to eddies of high angular momentum at small scale.

Migdal's Loop Equation and the Zeta Zeros

The connection between the zeta function and turbulence has been made even more explicit by Alexander Migdal. His loop equation program provides a rigorous derivation showing that the non-trivial zeros of the Riemann zeta function directly govern the complexity decay exponents of turbulence. Migdal's strategy is to abandon the local point-wise description of turbulence in favor of integrated loop observables, a move borrowed directly from quantum chromodynamics.

In 1993, Migdal predicted that the probability distribution of circulation Γ\Gamma Γ around a large loop CC C depends on the minimal surface area AA A bounded by the loop, and not on the loop's detailed shape. This is the direct turbulence analog of the Wilson loop area law in confining gauge theories, where the Wilson loop expectation decays with area, signaling quark confinement. The prediction went against every existing turbulence theory and was considered controversial. In 2019, a group at NYU confirmed it numerically via direct numerical simulations, finding that circulation is a bifractal quantity whose moments scale with loop area. This confirmation, twenty-six years after Migdal's original prediction, prompted him to return full-time to research in theoretical physics.

The area law also creates a structural parallel with the Ryu-Takayanagi formula from holographic gravity, where the entanglement entropy of a boundary region equals the area of the minimal bulk surface. All three contexts (QCD confinement, turbulence circulation, and holographic entropy) share a key physical observable computed from the area of a minimal surface bounded by specific boundary data. The shared minimal surface principle is, I think, the deepest geometric thread tying Migdal's turbulence to quantum gravity.

The most striking result of the Migdal program is how the non-trivial zeros of the Riemann zeta function appear as complex decay exponents of turbulence. This is not an assumption or a conjecture overlaid on the result. It emerges through a specific traceable chain of mathematical steps. The Euler ensemble involves star polygons with a co-primality constraint, and that constraint injects the Euler totient function into the statistical mechanics. When one computes the Mellin transform of the energy spectrum and averages over the ensemble, a Dirichlet series falls out. The physical consequence is log-periodic oscillations in turbulent correlation functions, oscillations on a logarithmic time scale generated by the imaginary parts of the zeta zeros. No phenomenological model of turbulence predicts them. Migdal reports emerging experimental evidence for them in high-precision wind tunnel data from the Max Planck turbulence group.

This result does not prove the Riemann hypothesis. But it creates a physical context in which the hypothesis has observable consequences. If all non-trivial zeros lie on the critical line, the complex exponents have a specific universal structure. An off-line zero would produce a detectable anomaly in turbulent decay measurements. The geometry of the Riemann zeta function becomes, in principle, measurable by experiment.

Cascades, Conformal Symmetry, and the Saturation Picture

The connection between the Majorana tower and turbulent cascades is currently at the level of conjecture rather than proven physical equivalence. But the parallels are detailed enough to constitute a serious research program.

In classical turbulence, energy cascades from large eddies to small ones through a self-similar sequence of vortex breakups, each scale contributing to the overall energy spectrum. The intermediate inertial range is characterized by scale invariance: the same physics repeating at every scale. This is precisely the structure described by conformal field theories, the quantum field theories of critical points. The Majorana tower provides infinite-dimensional unitary representations of the Lorentz group, which contains the conformal group at the appropriate limit. The tower's infinite spectrum, with its mass-spin relation, creates a level spacing that maps onto the cascade's energy levels. And periodic driving of Majorana-like systems to high angular momentum states, climbing the tower, produces spectral statistics governed by the zeros of the Riemann zeta function. That ties the tower directly to the turbulence decay exponents.

The picture I keep coming back to is that the critical line of the Riemann zeta function is the statistical saturation point of these systems. It governs the tipping points in models of macroscopic quantum-like behavior, including quantum chaos and fluid turbulence. The Migdal program and the Tamburini results, taken together, suggest that both turbulence and the Majorana tower may realize precisely such a system, and that they may be the same system seen through different lenses.

The physics here looks, to me, similar to the physics of dendritic arborization in brain tissue, or to magnetohydrodynamic instabilities in plasmas, since magnetohydrodynamics is essentially the same as ordinary hydrodynamics with the addition of Maxwell's equations. One way to see it is through Tesla coil streamers. Vacuum tube driven Tesla coils produce fractal streamers that seem to lack magnetohydrodynamic instabilities or streamer bifurcations. Streamer geometry can be modulated, with degrees of freedom pruned to a laminar setting or increased, in a way that resembles how dendritic connections are modulated in long-term depression and long-term potentiation. At the phase transition, there is a conformal fractal pattern across scales.

Migdal's 2025 extension to magnetohydrodynamic turbulence uses two coupled Euler ensembles, one for hydrodynamic circulation and one for magnetic circulation. The same zeta-zero structure persists, with an additional phase transition at magnetic Prandtl number Pm=1P_m = 1 Pm​=1 where the solution bifurcates. This robustness of the zeta structure across pure hydrodynamic and magnetohydrodynamic contexts strengthens the case that the zeta zeros represent a genuine universality class of cascade dynamics, and not an artifact of any one model.

Closing

Two Millennium Prize problems. One geometric structure. The Riemann zeta function turning up, again and again, at the saturation point of nonlinear deterministic systems, in the spectra of accelerated Majorana fields, in the loop statistics of turbulent fluids, and possibly in the bosonized output of driven entangled spin systems carrying orbital angular momentum. If this is right, then turbulence in a tokamak, the distribution of primes, the cascade of vortices in a wind tunnel, and the eigenvalues of the operator we have been searching for since Hilbert and Pólya are pieces of a single picture.

I do not claim this is settled. I claim it is the most interesting research direction I know of, and one where the experimental knobs are getting close enough to turn.