A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

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 a parts per thousand yen 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 psi(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 psi(x, y) for 5 a parts per thousand currency sign y < 17, compared with the result obtained by Ennola.