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This research aims to establish a conditional proof of the Riemann hypothesis using the Selberg zeta function.

July 28, 2026Open Access

A Conditional Proof of the Riemann Hypothesis via the Prime Geodesic Theorem, Quantum Unique Ergodicity, and Subconvexity

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This research aims to establish a conditional proof of the Riemann hypothesis using the Selberg zeta function.
Applied subconvexity bounds for GL2 L-functions to derive quantum unique ergodicity for geodesic flow on modular surface.
Proved a bridge lemma showing QUE leads to a specific bound on phase sums over Laplacian spectral parameters.
Substituted the bound into Hejhal's formula for the prime geodesic theorem to derive an error term.
Established a uniform O(T^{1/2+ε}) bound on phase sums via QUE.
Derived the error term πΓ(x) = li(x) + O(x^{1/2+ε}) from Hejhal's explicit formula.
Demonstrated that this error term supports the classical Riemann hypothesis, given the unconditional satisfaction of the Selberg zeta function.

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Conditional proof establishes Riemann hypothesis for Selberg zeta function, suggesting broader implications for mathematics.

xinlong he (2026) studied this question.

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