The Riemann Hypothesis, posited by Bernhard Riemann in 1859, has stood as one of the most important unsolved problems in mathematics for over 150 years. Its resolution has profound implications for our understanding of the distribution of prime numbers and touches on numerous areas of mathematics and theoretical physics. Concurrent with efforts to prove the Riemann Hypothesis, mathematicians have long sought to understand the nature of the zeta zeros through various frameworks, one of the most tantalizing being the Hilbert-Pólya Conjecture.

Spectral Realization of the Hilbert-Pólya Conjecture: A Novel Approach to the Riemann Hypothesis by G.O. Langford [Download]

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This paper presents a proof of the Riemann Hypothesis through a novel spectral approach, first realizing the Hilbert-Pólya Conjecture. We construct a self-adjoint operator \(A\_TN\) on a carefully defined Hilbert space \(H\_TN\), establishing a one-to-one correspondence between its eigenvalues and the non-trivial zeros of the Riemann zeta function. Our approach leverages a sophisticated interplay between functional analysis, complex analysis, and analytic number theory.

We introduce a function \(h(w)\) that serves as a bridge between the spectral properties of A_TN and the analytic properties of the Riemann zeta function. Through a series of theorems, we demonstrate that the eigenvalues of \(A\_TN\) correspond precisely to points on the critical line, thereby proving that all non-trivial zeros of the Riemann zeta function lie on the line \(Re(s) = 1/2\).

This work not only resolves one of the most famous open problems in mathematics but also provides a concrete realization of the long-hypothesized connection between zeta zeros and spectral theory. Our methodology, combining rigorous mathematical analysis with innovative conceptual frameworks, opens new avenues for tackling other significant problems in mathematics and related fields.

The implications of this result extend beyond number theory, potentially impacting areas such as quantum chaos, cryptography, and our understanding of prime number distributions. This paper represents a significant advancement in our comprehension of the deep structures underlying the Riemann zeta function and demonstrates the power of interdisciplinary approaches in modern mathematics.