  We extend the notion of generalised Cesaro summation/convergence developed
previously to the more natural setting of what we call "remainder" Cesaro
summation/convergence and, after illustrating the utility of this approach in
deriving certain classical results, use it to develop a notion of generalised
root identities. These extend elementary root identities for polynomials both
to more general functions and to a family of identities parametrised by a
complex parameter \mu. In so doing they equate one expression (the derivative
side) which is defined via Fourier theory, with another (the root side) which
is defined via remainder Cesaro summation. For \mu a non-positive integer these
identities are naturally adapted to investigating the asymptotic behaviour of
the given function and the geometric distribution of its roots. For the Gamma
function we show that it satisfies the generalised root identities and use them
to constructively deduce Stirling's theorem. For the Riemann zeta function the
implications of the generalised root identities for \mu=0,-1 and -2 are
explored in detail; in the case of \mu=-2 a symmetry of the non-trivial roots
is broken and allows us to conclude, after detailed computation, that the
Riemann hypothesis must be false. In light of this, some final direct
discussion is given of areas where the arguments used throughout the paper are
deficient in rigour and require more detailed justification. The conclusion of
section 1 gives guidance on the most direct route through the paper to the
claim regarding the Riemann hypothesis.

Number Theory;Classical Analysis and ODEs