Noncommutative Geometry, Modular Theory, and the Riemann Hypothesis

We study the distribution of the zeros of the Riemann zeta function within the noncommutative geometric framework introduced by A. Connes. Let X = A_Q/Q* denote the adele class space and let D be the infinitesimal generator of its natural scaling action. Fixing a distinguished faithful normal semifinite weight phi_0 in the adelic framework, we identify its modular automorphism group with the geometric scaling group via standard modular theory. Tomita-Takesaki theory yields D = log Deltaphi_0, whence D is self-adjoint. Combining the Connes trace formula, the separating property of the admissible class of test functions, and the real-support constraint on the spectral measure, we show that the self-adjointness of D forces every nontrivial zero of zeta(s) to lie on the critical line Re(s) = 1/2. Conversely, assuming the Riemann hypothesis, D is unitarily equivalent to a multiplication operator whose real spectrum consists of the ordinates of the nontrivial zeros; hence D is self-adjoint. Thus, within the Connes framework, the Riemann hypothesis is equivalent to the self-adjointness of the scaling generator D.