Yau, Stephen S.-T.; Yuan, Beihui; Zuo, Huaiqing On the polynomial sharp upper estimate conjecture in 7-dimensional simplex. (English) Zbl 1396.11119 J. Number Theory 160, 254-286 (2016). Summary: Because of its importance in number theory and singularity theory, the problem of finding a polynomial sharp upper estimate of the number of positive integral points in an \(n\)-dimensional (\(n \geq 3\)) polyhedron has received attention by a lot of mathematicians. The first named author proposed the Number Theoretic Conjecture for the upper estimate. The previous results on the Number Theoretic Conjecture in low dimension cases (\(n < 7\)) are proved by using the sharp GLY conjecture which is true only for low dimensional case. Thus the proof cannot be generalized to high dimension. In this paper, we offer a uniform approach to prove the Number Theoretic Conjecture for all dimensions by simply using the induction method and the Yau-Zhang [S. T. Yau and L. Zhang, Math. Res. Lett. 13, No. 5-6, 911–921 (2006; Zbl 1185.11062)] estimates (see Lemmas 2.3-2.5). As a result, the Number Theoretic Conjecture is proven for \(n = 7\). An important estimate for all dimensions is also obtained (Prepositions 3.1 and 3.2) which will be useful to prove the general case of the Number Theoretic Conjecture. As an application, we give a sharper estimate of the Dickman-De Bruijn function \(\Psi(x, y)\) for \(5 \leq y < 19\), compared with the result obtained by V. Ennola [Ann. Acad. Sci. Fenn., Ser. A I 440, 16 p. (1969; Zbl 0174.33903)]. Cited in 1 ReviewCited in 1 Document MSC: 11P21 Lattice points in specified regions 11Y99 Computational number theory Keywords:sharp estimate; integral points; simplex Citations:Zbl 1185.11062; Zbl 0174.33903 PDFBibTeX XMLCite \textit{S. S. T. Yau} et al., J. Number Theory 160, 254--286 (2016; Zbl 1396.11119) Full Text: DOI References: [1] Chen, I.; Lin, K.-P.; Yau, S. S.-T.; Zuo, H. 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