Hamiltonian with Energy Levels Corresponding to Riemann Zeros

Xingpao Suo

doi:10.1103/n9ch-xhw6

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arXiv · 2505.21192

Hamiltonian with Energy Levels Corresponding to Riemann Zeros

Xingpao Suo

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Abstract

A Hamiltonian with eigenenergy $ E_n = \rho_n(1 - \rho_n) $ has been constructed, where $ \rho_n $ denotes the $ n $-th non-trivial zero of the Riemann zeta function. To construct such a Hamiltonian, we generalize the Berry-Keating paradigm and encode number-theoretic information into the Hamiltonian using modular forms.Although our construction does not resolve the Hilbert-P\'olya conjecture (since the eigenstates corresponding to $ E_n $ are \emph{not} normalizable), it provides a novel physical perspective on the Riemann Hypothesis (RH). In particular, we propose a physical interpretation of RH, which could offer a potential pathway toward its proof.

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Xingpao Suo. 2025-05-27. Hamiltonian with Energy Levels Corresponding to Riemann Zeros. https://doi.org/10.1103/n9ch-xhw6

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