One can simulate harmonic analysis on a Riemannian manifold by replacing the de Rham complex with the cochain complex of a traingulated manifold, following the ideas that go back at least to A. Whitney and D. Sullivan, and more recently to S. Wilson. As a result, on obtains the discrete Hodge star operator acting on simplical cochains, which is an analogue of the usual Hodge star operator acting on differential forms. We will show that the discrete Hodge star operator is a topological invariant a 3-manifold; its action on the cohomology is determined by the underlying manifold together with its orientation, independently of the choice of a triangulation. We will discuss the relevance of this piece of combinatorial topology in number theory through the eye of the conjecture of Prasanna-Venkatesh.