The Riemann Hypothesis (RH) asserts that the nontrivial zeros of the Riemann zeta function lie on the critical line. This survey follows one research program—Alain Connes's approach to RH through noncommutative geometry—across ten papers (1998–2026). Its unifying idea is to realize the zeros as a spectrum on the adele class space X = A/k^*, under the scaling action of the idele class group, and to read the Weil explicit formula as a Lefschetz-type trace formula, thereby making RH equivalent to a positivity, namely the validity of a global trace formula. Two foundational works (Connes 1998; Connes–Consani–Marcolli 2007) realize the critical zeros as an absorption spectrum, recast the explicit formula on cyclic cohomology, and read the scaling action as a Frobenius in characteristic zero. The operator-theoretic (prolate) strand then replaces the Sobolev cutoff by geometric objects—the -cycle, the Sonin space, the prolate wave operator—turns Weil positivity into a spectral inequality, and yields a numerically verified spectral realization of the zeros, its self-adjointness supplied by an extension of the Carathéodory–Fejér theorem. The geometric (F₁) strand builds the arithmetic Jacobian and Picard objects of Spec\, Z, a translation–Lefschetz picture, and an absolute curve (Spec\, Z) ₅䃑 whose points unify the Fargues–Fontaine curve, the complex Tate curve and the scaling site. Throughout, the assessment is uniform: the framework is in place, but the decisive positivity mechanism—Riemann–Roch, or Hodge–Riemann—is still missing.
Guo Chen (Sun,) studied this question.