Counting an infinite number of points: a testing ground for renormalization methods

TL;DR

This paper explores how physics-inspired renormalization methods can be used to count infinite points, such as integer points on cones, through algebraic and coalgebraic frameworks, with applications to conical zeta functions.

Contribution

It introduces a coalgebraic approach to counting points on cones and applies renormalization techniques to conical zeta values, extending classical counting methods.

Findings

Renormalization methods can be applied to count points on cones.

A coalgebraic structure for cones is developed.

Conical zeta functions evaluate to counts of integer points.

Abstract

This is a leisurely introductory account addressed to non-experts and based on previous work by the authors, on how methods borrowed from physics can be used to "count" an infinite number of points. We begin with the classical case of counting integer points on the non-negative real axis and the classical Euler-Maclaurin formula. As an intermediate stage, we count integer points on product cones where the roles played by the coalgebra and the algebraic Birkhoff factorization can be appreciated in a relatively simple setting. We then consider the general case of (lattice) cones for which we introduce a conilpotent coalgebra of cones, with applications to renormalization of conical zeta values. When evaluated at zero arguments conical zeta functions indeed "count" integer points on cones.