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

We present a conditional proof of the Riemann hypothesis for the Selberg zeta function on the modular surface, which implies the classical Riemann hypothesis via the standard equivalence. Our proof proceeds in three steps. First, we apply recent subconvexity bounds for GL2 L-functions (Yang 2026; Miao-Zhang 2025; Hu-Petrow-Young 2026) to establish a quantitative form of quantum unique ergodicity (QUE) for the geodesic flow on the modular surface, following the framework of Nelson (2025). Second, we prove a bridge lemma showing that QUE implies a uniform O(T^{1/2+ε}) bound on the phase sums over the Laplacian spectral parameters. Third, we substitute this bound into Hejhal's explicit formula for the prime geodesic theorem, yielding the sharp error term πΓ(x) = li(x) + O(x^{1/2+ε}). Since the Selberg zeta function satisfies the Riemann hypothesis unconditionally, this error term is equivalent to the classical Riemann hypothesis. Thus, assuming the subconvexity results cited above, the Riemann hypothesis follows.