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

Key Points

• 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