Operators and Divergent Series

We give a natural extension of the classical definition of Césaro convergence of a divergent sequence/function. This involves understanding the spectrum of eigenvalues and eigenvectors of a certain Césaro operator on a suitable space of functions or sequences. The essential idea is applicable in identical fashion to other summation methods such as Borel’s. As an example we show how to obtain the analytic continuation of the Riemann zeta function ζ(z) for Re z ≤ 1 directly from generalised Césaro summation of its divergent defining series. We discuss a variety of analytic and symmetry properties of these generalised methods and some possible further applications.