A Spectral-Geometric Proof of the Riemann Hypothesis
9 days ago
- #Spectral Geometry
- #Analytic Number Theory
- #Riemann Hypothesis
- Presents a spectral-geometric proof of the Riemann Hypothesis, unifying analytic, operator-theoretic, and arithmetic formulations.
- Constructs a self-adjoint Sturm–Liouville operator whose spectrum corresponds bijectively to the nontrivial zeros of ζ(s).
- Uses Weyl–Titchmarsh and Herglotz frameworks to translate differential structure into analytic form, ensuring real eigenvalues.
- Employs Bochner integral and Paley–Wiener transform to derive a summation formula equivalent to Selberg’s trace formula.
- Certifies stability via Hilbert–Schmidt and Schur–Young estimates, ensuring spectral deformations remain continuous and measurable.
- Confines spectral counting function to discrete lattice S=mlogp, proving only atomic prime-power contributions survive.
- Proves uniqueness of arithmetic weights via Carlson’s theorem, fixing von Mangoldt weights as the only compatible coefficients.
- Eliminates classical escape routes for the Riemann Hypothesis to fail, ensuring self-adjointness, compactness, and symmetry.