n 維非整數座標頂點（非格頂點）多面體之格點數之估算為一重要問題。此估算可應用於數論之質數檢定與質因數分解及幾何與奇異理論之有趣應用。我們於2003 年提出GLY 猜想此猜想給出非格頂點n 維多面體之格點數之精確上估算。但是GLY 猜想要求n 維多面體（n 大於或等於3）每頂點離原點至少n-1。一個自然問題為如何提出無頂點離原點最少距離假設下之格點數之精確估算。本論文提出與Yau 幾何猜想直接對照的一個數論猜想。此論文給一般修正猜想之證明希望。做為一應用我們給Dickman-De Bruijn 函數ψ 於 小於11 時之精確估算。

Counting the number of integral points in n-dimensional tetrahedra with non-integral vertice is an important problem. It has applications in primality testing and factoring in number theory and interesting applications in geometry and singularity theory. We proposed GLY conjecture on sharp upper estimate of the number of integral points in n-dimensional tetrahedra with non-integral vertice in 2003. But GLY conjecture claim that the n dimensional (n ≥ 3) real right-angled simplice 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 a Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of Dickman-De Bruijn function for < 11.