  For 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 \lambda,
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(\lambda) and discuss when lambda \to
\infty in what cases N(lambda)=K(1/lambda).
