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

### Ke-Pao Lin

Chang Gung Institute of Technology, Tao-Yuan, Taiwan### Xue Luo

University of Illinois at Chicago, USA### Stephen S.-T. Yau

University of Illinois at Chicago, United States### Huaiqing Zuo

University of Illinois at Chicago, USA

## 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 $\leq x$ 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 ($n\geq3$) 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 $5\leq y<13$.

## Cite this article

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo, On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function. J. Eur. Math. Soc. 16 (2014), no. 9, pp. 1937–1966

DOI 10.4171/JEMS/480