On Weyl formula for the Laplace operator over the compact surfaces

The circle problem is reduced to the explicit estimate of the second term of Weyl formula for the Laplace operator on the two-dimensional torus. The talk is devoted to the analogous problems for the cases of non-planar metrics and for the cases of the surfaces whose genus is greater than one. The survey of the present results and some hypothesis (including the hypothesis of quantum chaos) will be presented.