Discrete sums of classical symbols on Z^d and zeta functions associated with Laplacians on tori

TL;DR

This paper develops a canonical regularization for discrete sums of classical symbols on Z^d, enabling meromorphic extensions of sums and zeta functions related to Laplacians on tori, with explicit residue formulas.

Contribution

It introduces a unique translation-invariant extension of discrete sums for classical symbols, generalizing previous results and applying them to zeta functions of quadratic forms and Laplacians.

Findings

Established a canonical regularized sum for classical symbols on Z^d.

Derived explicit formulas for residues of associated zeta functions.

Extended previous work to include Laplacians on tori.

Abstract

We prove the uniqueness of a translation invariant extension to non integer order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we then describe using an Hadamard finite part procedure for sums over integer points of infinite unions of nested convex polytopes in $\R^d$. This canonical regularised sum is the building block to construct meromorphic extensions of the ordinary sum on holomorphic symbols. Explicit formulae for the complex residues at their poles are given in terms of noncommutative residues of classical symbols, thus extending results of Guillemin, Sternberg and Weitsman. These formulae are then applied to zeta functions associated with quadratic forms and with Laplacians on tori.