Renormalised conical zeta values

TL;DR

This paper introduces a renormalisation method for conical zeta values associated with rational convex polyhedral cones, extending algebraic techniques to handle poles and deriving an Euler-Maclaurin formula for lattice cones.

Contribution

It generalises Connes and Kreimer's algebraic renormalisation to conical zeta values, providing new tools for their analysis and applications to lattice cones.

Findings

Developed a renormalisation scheme for conical zeta values at poles.

Derived an Euler-Maclaurin formula relating sums and integrals on lattice cones.

Connected the scheme to existing algebraic structures like the Birkhoff decomposition.

Abstract

Conical zeta values associated with rational convex polyhedral cones generalise multiple zeta values. We renormalise conical zeta values at poles by means of a generalisation of Connes and Kreimer's Algebraic Birkhoff Factorisation. This paper serves as a motivation for and an application of this generalised renormalisation scheme. The latter also yields an Euler-Maclaurin formula on rational convex polyhedral lattice cones which relates exponential sums to exponential integrals. When restricted to Chen cones, it reduces to Connes and Kreimer's Algebraic Birkhoff Decomposition for maps with values in the algebra of ordinary meromorphic functions in one variable.