  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.
