The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator
- 1Independent Researcher
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.
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BibTeX
@misc{chen2026riemann,
title = {{The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator}},
author = {Zhuo Chen},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.22074403},
url = {https://doi.org/10.5281/zenodo.22074403},
note = {CC BY},
}
RIS
TY - UNPB
TI - The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator
AU - Zhuo Chen
PY - 2026
DA - 2026/08/24
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22074403
UR - https://doi.org/10.5281/zenodo.22074403
AB - 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.
ER -