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

Abstract

It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $\psi (x,y)$ which is the number of positive integers $\le x$ and free of prime factors $>y$. Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ($n\ge 3$) real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for $n=5$. As an application, we give a sharp estimate of Dickman-De Bruijn function $\psi (x,y)$ for $5\le y<13$.