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This research introduces a novel framework for analyzing the Riemann zeta function using quadratic spectral operators.

May 7, 2026Open Access

ZPIF (Zero Pair Interaction Functional): A Quadratic Spectral Operator Framework with Heuristic Connections to the Riemann Explicit Formula — Analytical and Computational Perspectives

Key Points

This research introduces a novel framework for analyzing the Riemann zeta function using quadratic spectral operators.
Developed ZPIF framework incorporating second-order interactions in Hilbert space.
Established rigorous definitions, including spectral expansion and trace-class regularization.
Proposed a computational scheme based on numerical zeta zeros.
Demonstrated nonlinear growth behavior not present in classical formulations.
Observed quadratic interaction effects indicating advancement over standard linear spectral methods.

Abstract

ZPIF (Zero Pair Interaction Functional) is introduced as a quadratic spectral operator framework extending the classical explicit formula of the Riemann zeta function. Unlike the standard linear spectral decomposition, ZPIF incorporates second-order interactions between spectral modes within a Hilbert space formulation. The framework includes a rigorous operator definition, spectral expansion, trace-class regularization, and conditional convergence under truncation. A computational scheme based on numerical zeta zeros is also proposed. The novelty of ZPIF lies in introducing a quadratic spectral energy functional consistent with classical spectral heuristics without assuming unresolved conjectures. Numerical experiments demonstrate nonlinear growth behavior and quadratic interaction effects that are absent in classical linear formulations.

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