JournalsjemsVol. 16, No. 9pp. 1937–1966

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

  • Ke-Pao Lin

    Chang Gung Institute of Technology, Tao-Yuan, Taiwan
  • Xue Luo

    University of Illinois at Chicago, USA
  • Stephen S.-T. Yau

    University of Illinois at Chicago, United States
  • Huaiqing Zuo

    University of Illinois at Chicago, USA
On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function cover
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Abstract

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)\psi(x,y) which is the number of positive integers x\leq x and free of prime factors >y>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 (n3n\geq3) real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for n=5n=5. As an application, we give a sharp estimate of Dickman-De Bruijn function ψ(x,y)\psi(x,y) for 5y<135\leq y<13.

Cite this article

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. 1937–1966

DOI 10.4171/JEMS/480