On the polynomial sharp upper estimate conjecture in 7-dimensional simplex

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≥3n≥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<7n<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 [19] estimates (see Lemma 2.3, Lemma 2.4 and Lemma 2.5). As a result, the Number Theoretic Conjecture is proven for n=7n=7. An important estimate for all dimensions is also obtained ( Proposition 3.1 and Proposition 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 ψ(x,y)ψ(x,y) for 5≤y<195≤y<19, compared with the result obtained by Ennola.