2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154886For a given spectrum {lambda_{n}} of the Laplace operator on a Riemannian manifold, in this paper, we present a relation between the counting function N(lambda), the number of eigenvalues (with multiplicity) smaller than λ, and the heat kernel K(t), defined by K(t)=\sum_{n}e^{-lambda_{n}t}. Moreover, we also give an asymptotic formula for N(λ) and discuss when lambda \to \infty in what cases N(lambda)=K(1/lambda).5 pages, no figureSpectral Theorytext