A sharp polynomial estimate of positive integral points in a 4-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

The estimate of integral points in right-angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional (n≥3) real right-angled simplices with vertices whose distance to the origin are at least n-1. A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for n=4. This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman-de Bruijn function ψ(x,y) for y<11.