Generalised Cesaro Convergence, Root Identities and the Riemann Hypothesis

Abstract

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.