The estimate of integral points in an n-dimensional polyhedron has many applications
in singularity theory, number theory and toric geometry. The third named author
formulated a conjecture (i.e., Yau Number Theoretic Conjecture) which gives a sharp polynomial
upper estimate on the number of positive integral points in n-dimensional (n 3)
real right-angled simplices. The previous results on the conjecture in low dimension cases
(n 6) have been proved by using the sharp GLY conjecture. However, it is only valid
in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with
a completely new method in [22]. In this paper, on the one hand, the similar method has
been applied to prove the 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
seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a
signi cant progress since it sheds light on proving the Yau Number Theoretic Conjecture
in full generality. On the other hand, we give a new sharper estimate of the Dickman-De
Bruijn function (x; y) for 5 y < 23, compared with the result obtained by Ennola.