  As well known, the important hypothesis formulated by B.G. RIEMANN in 1859
states that all non-trivial zeroes of the Zeta function
$Z(s)=\sum_{n=1}^{\infty } n^{-s}$ should fall on the Critical Line (C.L.)
$Re(s)=\frac{1}{2}$.\\ Although direct numerical search of the zeroes failed to
identify any outlier, i.e. any zeroes with $Re(s)\neq\frac{1}{2}$, a general
proof of the Hypothesis has not yet been found.\\ The present Note aims to
approach the problem from a 'reductio ad absurdum' way, i.e. it assumes that an
outlier pair of c.c. zero-points, $s=\frac{1}{2}+\xi \pm i.Y_H$ with
$\xi\neq0$, has been found, and then proceeds to analyze what are the
implications of this assumption. Starting from the well-known GRAM-BACKLUND
formula for an explicit expression of the Zeta function, the Fundamental
Theorem of Algebra (FTA) allows to evidence, through legitimate algebraic
manipulations, the necessity that the assumed outlying pair of c.c. zero-points
fulfils an implicit additional constraint. The zero-condition according to the
GRAM- BACKLUND formulation and this additional constraint are seen to be
mutually incompatible unless the pair of c.c. zero-points belong to the C.L..
This conclusion is equivalent to a verification of the RIEMANN Hypothesis.
