Journal of the European Mathematical Society


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Volume 16, Issue 9, 2014, pp. 1937–1966
DOI: 10.4171/JEMS/480

Published online: 2014-10-22

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[1], Xue Luo[2], Stephen S.-T. Yau[3] and Huaiqing Zuo[4]

(1) Chang Gung Institute of Technology, Tao-Yuan, Taiwan
(2) University of Illinois at Chicago, USA
(3) University of Illinois at Chicago, United States
(4) University of Illinois at Chicago, USA

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 $5\leq y<13$.

Keywords: Tetrahedron, Yau number-theoretic conjecture, upper estimate

Lin Ke-Pao, Luo Xue, Yau Stephen, Zuo Huaiqing: 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), 1937-1966. doi: 10.4171/JEMS/480