Average growth of the spectral function on a Riemannian manifold

TL;DR

This paper investigates the average growth behavior of the spectral function of the Laplacian on Riemannian manifolds, considering different averaging methods and related zeta-function estimates, with discussions on almost periodic properties.

Contribution

It introduces new average growth estimates for the spectral function and explores related zeta-function behavior and open problems in spectral theory on manifolds.

Findings

Derived growth estimates for spectral functions under various averaging schemes

Established bounds for the pointwise zeta-function along vertical lines

Discussed open problems related to almost periodicity of spectral functions

Abstract

We study average growth of the spectral function of the Laplacian on a Riemannian manifold. Two types of averaging are considered: with respect to the spectral parameter and with respect to a point on a manifold. We obtain as well related estimates of the growth of the pointwise zeta-function along vertical lines in the complex plane. Some examples and open problems regarding almost periodic properties of the spectral function are also discussed.