On a number-theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

It is well known that getting an estimate of the number of integral points in right-angled simplices is equivalent to getting an estimate of the Dickman-de Bruijn function psi(x, y) which is the number of positive integers <= x and free of prime factors > y. Motivated by the Yau Geometric Conjecture, the third author formulated a 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. In this paper, we prove this conjecture for n = 5. As an application, we give a sharp estimate of the Dickman-de Bruijn function psi(x, y) for 5 <= y < 13.