The spectral function Θ ( t ) = ∑ i = 1 ∞ exp ( − t λ j ) , where { λ j } j = 1 ∞ are the eigenvalues of the negative Laplace-Beltrami operator − Δ , is studied for a compact Riemannian manifold Ω of dimension “ k ” with a smooth boundary ∂ Ω , where a finite number of piecewise impedance boundary conditions ( ∂ ∂ n i + γ i ) u = 0 on the parts ∂ Ω i ( i = 1 , … , m ) of the boundary ∂ Ω can be considered, such that ∂ Ω = ∪ i = 1 m ∂ Ω i , and γ i ( i = 1 , … , m ) are assumed to be smooth functions which are not strictly positive.