Geodesic
On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function
Journal of the European Mathematical Society, Tome 16 (2014) no. 9, pp. 1937-1966.

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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 ψ(x,y) which is the number of positive integers ≤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≥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 ψ(x,y) for 5≤y13.
DOI : 10.4171/jems/480
Classification : 11-XX, 00-XX
Keywords: Tetrahedron, Yau number-theoretic conjecture, upper estimate 
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     title = {On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the {Dickman{\textendash}De} {Bruijn} function},
     journal = {Journal of the European Mathematical Society},
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     doi = {10.4171/jems/480},
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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. Journal of the European Mathematical Society, Tome 16 (2014) no. 9, pp. 1937-1966. doi : 10.4171/jems/480. http://geodesic.mathdoc.fr/articles/10.4171/jems/480/

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