Introduction to generalised Cesaro convergence I

TL;DR

This paper introduces generalized Cesaro convergence methods that extend classical convergence, enabling the summation of divergent series and functions, with applications in complex analysis and number theory.

Contribution

It presents a new framework for generalized Cesaro convergence, broadening the scope of summation methods for divergent sequences and series.

Findings

Provides a constructive method for analytic continuation of complex functions.

Extends Cesaro methods to a wider class of divergent series.

Lays groundwork for future research in number theory and analysis.

Abstract

This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of limits/sums for a much broader class of divergent sequences/series. These provide a constructive means of analytic continuation of functions of a complex variable and we give many examples. Future sets of papers will use these methods to derive new results (and re-derive many existing results) in areas including analytic number theory; the theory of the Riemann zeta function; reversal of order of summation; exponential sums; classical integration; Taylor series and Mellin transforms; asymptotic analysis; and a number of others.