What question did this study set out to answer?

To propose a spectral construction aligned with the Riemann Hypothesis using a self-adjoint operator.

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March 31, 2026Open Access

A Spectral Construction Toward the Riemann Hypothesis

Key Points

To propose a spectral construction aligned with the Riemann Hypothesis using a self-adjoint operator.
Defined a self-adjoint operator replicating the imaginary parts of Riemann zeta function zeros.
Combined inverse spectral theory with the classical explicit formula.
Established a trace relation related to prime number distribution.
Implemented a positivity condition following Weil’s criterion.
Developed an operator-theoretic structure consistent with the Riemann Hypothesis.
Showed that the spectrum of the operator aligns with non-trivial zeros of the zeta function.
Demonstrated a coherent relationship between trace relations and prime distribution.

Abstract

This short note presents a constructive spectral framework related to the Hilbert–Pólya program.A self-adjoint operator is defined whose spectrum reproduces the imaginary parts of the non-trivial zeros of the Riemann zeta function. The construction combines inverse spectral theory with the classical explicit formula, yielding: a self-adjoint operator with prescribed spectrum, a trace relation consistent with prime number distribution, and a positivity condition aligned with Weil’s criterion. Together, these elements form a coherent operator-theoretic structure consistent with the Riemann Hypothesis.

A Spectral Construction Toward the Riemann Hypothesis

Key Points

Abstract

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