How (not) to prove the Riemann hypothesis with quantum mechanics
Summary:
We will start with a roundtable discussion of this meme. One approach to the Riemann hypothesis is to find an operator whose eigenvalues are the zeros of the Riemann zeta function, then prove that operator is self-adjoint (The Hilbert-Ploya conjecture). I will tell you about a proposed way to construct such operators using symplectic geometry and quantization. This leaves us with a treasure hunt, a search for a hypothetical hamiltonian whose quantization solves the Riemann hypothesis. No one before, as far as I can tell, has used the edifice of contact homology to try to find this hamiltonian. Maybe there’s something we can do…
Presented at:
- Student Symplectic Summer Sessions, Summer 2025
🔗 Link to file
Video of talk
Sources
- RIEMANN’S ZETA FUNCTION: A MODEL FOR QUANTUM CHAOS? by Berry
- Berry’s origional proposal that the Riemann explicit formula has a lot of similarities with trace formulae from the field of quantum chaos.
- H=xp AND THE RIEMANN ZEROS by Berry and Keating
- Proposes that a quantization of the hamiltonian $xp$ on a 2D phase space has the same asymptotic distribution of eigenvalues as the zeros of the Riemann Zeta function.
- A compact hamiltonian with the same asymptotic mean spectral density as the Riemann zeros by Berry and Keating
- A more recent followup of Berry and Keating’s original paper, with a well defined proposed Hamiltonian. This hamiltonian smooths out the issues with $xp$, while having the same semiclassical asymptotics. This doesn’t match the precise zeros of the Riemann Zeta function, though.
- Hearing the music of the primes: auditory complementarity and the siren song of zeta by Michael Berry
- A fun expository paper, making precise an analogy between primes and music. Not only that, he made the songs! You can listen to them!