We consider the set of distances from a point to a lattice in Euclidean space, for a metric related to a convex body. Associated with these lengths, we construct a Poincaré series: a natural holomorphic function defined in a complex half-plane. The aim of the talk is to study this function: its possible extension to the other half-plane, its poles, its singularities, etc. In doing so, we encounter a multiplication operator by a Morse function on the sphere and describe its spectral theory. This is joint work with Nguyen Viet Dang, Yannick Guedes-Bonthonneau, and Gabriel Rivière.
