  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.
