Surface of Revolution

We prove that the error term Λ(R) in the Weyl asymptotic formula #{En ≤ R } = VolM 4π R + Λ(R), for the Laplace operator on a surface of revolution M satisfying a twist hypothesis, has the form Λ(R) = RF (R) where F (R) is an almost periodic function of the Besicovitch class B, and the Fourier series of F (R) in B is ∑ γ A(γ) cos(|γ|R− φ) where the sum goes over all closed geodesics on M , and A(γ) is computed through simple geometric characteristics of γ. We extend this result to surfaces of revolution, which violate the twist hypothesis and satisfy a more general Diophantine hypothesis. In this case we prove that Λ(R) = RΦ(R) + RF (R), where Φ(R) is a finite sum of periodic functions and F (R) is an almost periodic function of the Besicovitch class B. The Fourier series of Φ(R) and F (R) are computed.