A synthetic algebraic approach to a discussion of Riemann's hypothesis

Abstract

A fresh approach to the long debated question is proposed, starting from the GRAM-BACKLUND analytical continuation of the Zeta function (G-B Zeta expression). Consideration is given to the symmetric (even-exponent) and anti-symmetric (odd exponent) components of the power series representation of the G-B formulation along a circular path containing the 4 hypothetical zero-points predicted by HADAMARD and DE LA VALLEE POUSSIN (H/DLVP 'outlying' quartet). From the necessary conditions required of the even- and odd-exponent components at a representative zero-point of the hypothetical quartet some interesting logical consequences are derived and briefly discussed in the framework of the RIEMANN Hypothesis. A temporary mapping of the representative zero-point of the hypothetical quartet onto a higher-dimensional auxiliary domain provides an intriguing short-cut to the negative conclusion about the possibility of existence of such outlying zero-points.