Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

TL;DR

This paper explores the conditions under which the spectral zeta function of the Laplacian on self-similar fractals can be extended meromorphically, combining classical and new probabilistic methods.

Contribution

It introduces new results on meromorphic extension of spectral zeta functions on fractals, integrating heat kernel estimates with existing theories.

Findings

Meromorphic extension exists for certain fractals

Heat kernel estimates are crucial for extension proofs

Formulation of conjectures for broader classes of fractals

Abstract

We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature.