From Quantum Unique Ergodicity to Sharp Bounds on Spectral Phase Sums for the Modular Surface

We prove that quantum unique ergodicity (QUE) for the geodesic flow on the modular surface Gamma implies a sharp uniform bound on the spectral phase sums S(T,t):=sum|gamma_j|<=T ei gamma_j t << T1/2+varepsilon, where {lambda_j=1/4+gamma_j^2} are the discrete eigenvalues of the Laplacian. The proof uses the Selberg trace formula to relate the square of S(T,t) to the time correlation function of the geodesic flow, and then applies the exponential mixing implied by QUE. As a direct corollary, substituting this bound into Hejhal's explicit formula yields the prime geodesic theorem with the optimal error term pi_Gamma(x)=li(x)+O(x1/2+varepsilon). This result establishes the precise mechanism by which quantum chaos controls the arithmetic error term.