The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator
Year: 2026
Type: preprint
Abstract: This article presents a complete proof of the Riemann hypothesis (RH). The approach proceeds through classical analytic methods: the Herglotz representation theorem, the theory of orthogonal polynomials on the real line, and the self-adjoint extension theory of Jacobi operators. We construct an explicit Hilbert-Pólya operator—a self-adjoint Jacobi operator whose discrete spectrum coincides with the squares of the imaginary parts of the non-trivial zeros of the Riemann zeta function. The proof verifies the key positivity and monotonicity conditions needed for Herglotz analyticity, completing the verification of RH. Version 4 (October 2026): Adds Appendix F with fully rigorous proofs of the angular-monotonicity numerical-verification gaps. All four CRITICAL gaps (C1–C4) and all four MAJOR gaps (M1–M4) are now resolved: the large-r asymptotic (gap C1), Subregion B (gap M2), Subregion C for r ∈ [1000, 75000] (gap C4), the max |f(w)| bound on |w| = 13 (gap M1), and zero-data sourcing (gap M4). The mathematical content of the main proof is unchanged from v3.
Author: Chen, Zhuo
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204092383