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 which is the number of positive integers and free of prime factors . 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 () real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for . As an application, we give a sharp estimate of Dickman-De Bruijn function for .
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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. 1937–1966DOI 10.4171/JEMS/480