A SPECTRAL CRITERION FOR THE ABSENCE OF SIEGEL ZEROS VIA REGULARIZED MAASS-SELBERG RELATIONS

This paper presents a conditional proof of the Riemann Hypothesis based on the spectral theory of the Laplacian on the modular surface X = PSL(2, Z) \ H. We establish a direct connection between the existence of Siegel zeros (real poles of the scattering coefficient) and the non-vanishing of certain regularized inner products. By employing the Maass-Selberg relation for truncated Eisenstein series and Zagier’s regularization method, we derive an explicit rational form for the interaction coefficient between the residual spectrum and the continuous spectrum. We demonstrate that the existence of a Siegel zero implies a meromorphic identity phi(z) = Q(z), where Q(z) is a rational function with a single pole, contradicting the known analytic structure of the scattering matrix phi(z).