Let Í2 be a compact Riemannian manifold of dimension n with smooth boundary. Let Aj < X2 < • ■ ■ be the eigenvalues of the Laplace-Beltrami operator with the boundary condition [d/dn + y] = 0 . The associated 8-function Qy(t) = ^2^=xe\p[-X"t] has an asymptotic expansion of the form (47tt)n/2ey(t) = a0 + axtxl2 + a2t + a3t^2 + aAt2 + ■■■ . The values of Oq , ax are well known. We compute the coefficients a% and a$ in terms of geometric invariants associated with the manifold by studying the

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... etrix expansion of the heat kernel p(t, x, y) near the boundary. Our method is a significant refinement and improvement of the method used in