Because of its importance in number theory and singularity theory, the problem of nding
a polynomial sharp upper estimate of the number of positive integral points in an n-
dimensional (n 3) polyhedron has received attention by a lot of mathematicians. S. S.-T.
Yau proposed the upper estimate, so-called the Yau Number Theoretic Conjecture. The previous
results on the Yau Number Theoretic Conjecture in low dimension cases (n 6) have
been proved by using the sharp GLY conjecture. Unfortunately, it is only valid in low dimension.
The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new
method in [19]. In this paper, the similar method has been applied to prove the Yau Number
Theoretic Conjecture for n = 8, but with more meticulous analyses. The main method of
proof is summing existing sharp upper bounds for the number of points in 7-dimensional
simplexes over the cross sections of eight-dimensional simplex. This reasearch project paves
the way for the proof of a fully general sharp upper bound for the number of lattice points
in a simplex. It also moves the mathematical community one step closer towards proving the
Yau Number Theoretic Conjecture in full generality. As an application, we give a sharper
estimate of the Dickman-De Bruijn function (x; y) for 5 y < 23, compared with the
result obtained by Ennola.