What question did this study set out to answer?

The research aims to connect the Brindel transformation with Fourier analysis, particularly focusing on the Riemann zeta function.

March 10, 2026Open Access

The Brindel Transformation and Fourier Analysis: A Rotation Identity and its Consequences

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The research aims to connect the Brindel transformation with Fourier analysis, particularly focusing on the Riemann zeta function.
Analyzed the relationship between the Brindel transformation and Fourier analysis.
Investigated the properties of the transformed Euler series.
Identified a rotation identity involving the factor e^{-2πσi}.
Examined conditions for the modulus |F(s)| = 1 and its implications.
Demonstrated that the transformed Euler series satisfies a rotation identity.
Found that the factor e^{-2πσi} equals -1 when σ = 1/2.
Connected the rotation factor with the modulus condition from the Brindel transformation.
Identified an open problem regarding the Cauchy-type singularity and its implications for off-line zeros.

Abstract

We establish a new connection between the Brindel transformation and the Fourier analysis of the Riemann zeta function. The central result is that the transformed Euler series satisfies the rotation identity. The factor e^-2πσi is a rotation in C that equals -1 if and only if σ = 1/2. We show that this rotation factor coincides with the modulus condition |F (s) | = 1 from the main Brindel transformation paper, providing a second characterization of the critical line via Fourier analysis. The precise open problem is identified: whether the Cauchy-type singularity arising from a hypothetical off-line zero is irremovable.

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Judicael Brindel (Sun,) studied this question.

synapsesocial.com/papers/69af956970916d39fea4cebf — DOI: https://doi.org/10.5281/zenodo.18907491

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The Brindel Transformation and Fourier Analysis: A Rotation Identity and its Consequences

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