Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture (see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional (n 3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5 y < 17, compared with the result obtained by Ennola.