On the GLY conjecture of upper estimate of positive integral points in real right-angled simplices (Q863967)

The authors study the number \(P_{ (a_1,\dots,a_n) } \) of integral points in the \(n\)-dimensional tetrahedron \[ \left\{ (x_1,\dots,x_n) \in {\mathbb R}_{>0}^n : \;{{x_1} \over {a_1}} + \cdots + {{x_n} \over {a_n}} \leq 1 \right\} , \] where \(a_1,\dots,a_n\) are positive real numbers. Lattice point problems in these tetrahedra have many applications, for example, in analytic number theory (primality testing and factoring) and singularity theory (Durfee conjecture). Formulas for specific cases, in which \(a_1,\dots,a_n\) are integers, have been given by \textit{L. J. Mordell} [ J. Indian Math. Soc. (N.S.) 15, 41--46 (1951; Zbl 0043.05101)], \textit{J. E. Pommersheim} [Math. Ann. 295, No. 1, 1--24 (1993; Zbl 0789.14043)], and \textit{J.-M. Kantor, A. Khovanskii} [C. R. Acad. Sci. Paris, Sér. I Math. 317, No. 5, 501--507 (1993; Zbl 0791.52012)]. In a series of papers [\textit{Y.-J. Xu, S. S.-T. Yau}, J. Reine Angew. Math. 423, 199--219 (1991; Zbl 0734.11048), J. Reine Angew. Math. 473, 1--23 (1996; Zbl 0844.11063)], and [\textit{K.-P. Lin, S. S.-T. Yau}, J. Number Theory 93, 207--234 (2002; Zbl 0992.11057)], Lin, Xu, and Yau established upper bounds for \(n=3\), 4, and 5. These results inspired the ``Granville-Lin-Yau conjecture,'' which consists of two parts: (1) If \(a_1 \geq a_2 \geq \cdots \geq a_n \geq n-1 \geq 2\), then \[ n! \, P_{ (a_1,\dots,a_n) } \leq f_n := A_0^n + {{ s(n,n-1) } \over { n }} A_1^n + \sum_{ k=1 }^{ n-2 } {{ s(n,n-1-k) } \over { \left( {{ n-k } \atop k} \right) }} A_k^{ n-1 } \] and \(n! \, P_{ (a_1,\dots,a_n) } = f_n\) if and only if \(a_1 = a_2 = \cdots = a_n\) is an integer. (2) If \(a_1 \geq a_2 \geq \cdots \geq a_n > 1\) and \(n \geq 3\), then \[ n! \, P_{ (a_1,\dots,a_n) } < \left( a_1 - 1 \right) \left( a_2 - 1 \right) \cdots \left( a_n - 1 \right) . \] Here \(s(n,k)\) denotes the Stirling number of the first kind and \[ A_k^n = \prod_{ i=1 }^n a_i \sum_{ 1 \leq i_1 < \dots < i_k \leq n } {1 \over { a_{ i_1 } \cdots a_{ i_k } }} \, . \] In the paper under review, the authors prove (2) for \(n=4, 5, 6\) and give a counterexample to (1) for \(n=7\). Thus the authors revise the first part and conjecture that (1) holds when \(a_1 \geq a_2 \geq \cdots \geq a_n \geq \alpha\) for some \(\alpha\) that only depend on \(n\). Subsequently, (2) was proved for general \(n\) by \textit{S. T. Yau} and \textit{L. Zhang} [Math. Res. Lett. 13, No. 5-6, 911--921 (2006; Zbl 1185.11062)].