Lattice Points in Cones and Dirichlet Series

Hecke proved the meromorphic continuation of a Dirichlet series associated to the lattice points in a triangle with a real quadratic slope and found the possible poles in terms of the fundamental unit. An analogous result is proven for certain elliptical cones where now the poles are determined by the spectrum of the Laplacian on an arithmetic Riemann surface. 1 Some results of Hardy-Littlewood and Hecke In the early 1920’s Hardy-Littlewood and Hecke studied the analytic properties of a Dirichlet series whose summatory function counts lattice points in a triangle. More precisely, for fixed q ∈ R, consider the triangle in the x, y-plane whose sides are given by qx = y and y = M. Letting a(m) = aq(m) = #{x ∈ Z | qx < m2} we clearly have that ∑ m<M a(m) is the number of lattice points in this triangle.