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This research investigates how certain linear operators affect the representation in the binary Goldbach problem.

May 19, 20260 citationsOpen Access

A Quantitative Structural Obstruction for Linear Diagonal Filters in the Binary Goldbach Problem

Key Points

This research investigates how certain linear operators affect the representation in the binary Goldbach problem.
Analyzed linear operators acting on the Fourier basis
Established the high-frequency decay condition for multipliers
Examined implications under the Bombieri–Vinogradov theorem and Generalized Riemann Hypothesis
Linear operators with high-frequency decay M_N = o((log N)^{−2}) annihilate the main term in Goldbach representations.
The findings highlight failures of linear filtering methods, necessitating bilinear approaches.
Results extend to general bounded operators through orthogonal projections.

Abstract

We establish a quantitative structural obstruction for linear operators acting diagonally on the Fourier basis in the context of the binary Goldbach problem. Any uniformly bounded linear operator TN whose multipliers satisfy the high-frequency decay condition MN = o ( (log N) ^−2) annihilates the main term of the weighted Goldbach representation function. The result extends to general bounded operators via orthogonal projections, providing a rigorous explanation for the failure of purely linear filtering methods and the structural indispensability of bilinear techniques. Conditional refinements under the Bombieri–Vinogradov theorem and the Generalized Riemann Hypothesis are discussed.

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