Meromorphic continuation approach to noncommutative geometry

TL;DR

This paper introduces a method based on generalized differential operators to establish meromorphic continuation of spectral zeta functions, with applications to nilpotent Lie algebras, advancing noncommutative geometry techniques.

Contribution

It develops a new approach for meromorphic continuation of spectral zeta functions using algebras of generalized differential operators, extending previous ideas in noncommutative geometry.

Findings

Proves existence of meromorphic continuation under specific conditions.

Locates poles within arithmetic sequences.

Provides bounds on pole orders.

Abstract

Following an idea of Nigel Higson, we develop a method for proving the existence of a meromor-phic continuation for some spectral zeta functions. The method is based on algebras of generalized differential operators. The main theorem states, under some conditions, the existence of a meromor-phic continuation, a localization of the poles in supports of arithmetic sequences and an upper bound of their order. We give an application in relation with a class of nilpotent Lie algebras.