What question did this study set out to answer?

The research aims to establish structural equivalences related to the Riemann Hypothesis using a new operator-theoretic framework.

January 20, 2026Open Access

Logarithmic-Potential Operator Frameworks and the Riemann Hypothesis: Structural Equivalences at the Critical Line

Key Points

The research aims to establish structural equivalences related to the Riemann Hypothesis using a new operator-theoretic framework.
Introduced a conditional operator-theoretic framework based on logarithmic potentials from the Riemann ξ-function.
Constructed two operator realizations: Friedrichs extension and natural realization.
Proved equivalence under four framework hypotheses.
Established equivalence L_F = L_N under the condition of RH.
Demonstrated that λ_n ≥ 0 is equivalent to this operator equality.
Showed spectral-nodal correspondence as part of the equivalence.

Abstract

We introduce a conditional operator-theoretic framework based on logarithmic potentials derived from the Riemann ξ-function at the critical line. Two operator realizations are constructed: 1. Friedrichs extension LF (via quadratic forms) 2. Natural realization LN (via Weyl m-function) Under four framework hypotheses, we prove the equivalence: RH ⟺ (LF = LN) ⟺ (λₙ ≥ 0) ⟺ (Spectral-Nodal Correspondence) SCOPE: This paper establishes rigorous equivalences within the conditional framework. It does NOT prove the Riemann Hypothesis, nor does it prove the natural realization LN exists unconditionally. MSC 2020: 11M26 (primary), 47E05, 47B25, 35P05 (secondary) Includes an elementary-level explanation guide for accessibility.

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