Lattice Points in the Four-Dimensional Ball
- 作者: Fomenko O.1
-
隶属关系:
- St. Petersburg Department of the Steklov Mathematical Institute
- 期: 卷 234, 编号 5 (2018)
- 页面: 750-757
- 栏目: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/241992
- DOI: https://doi.org/10.1007/s10958-018-4040-5
- ID: 241992
如何引用文章
详细
Let r4(n) denote the number of representations of n as a sum of four squares. The generating function ζ4(s) is Epstein’s zeta function. The paper considers the Riesz mean
\( {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_4(n) \)
for an arbitrary fixed ρ > 0. The error term Δρ(x; ζ4) is defined by
\( {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{\uppi^2{x}^{2+\rho }}{\Gamma \left(\rho +3\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_4(0)+{\Delta}_{\rho}\left(x;{\zeta}_4\right). \)
It is proved that
\( {\Delta}_4\left(x;{\zeta}_4\right)=\Big\{{\displaystyle \begin{array}{ll}O\left({x}^{1/2+\rho +\varepsilon}\right)& \left(1<\rho \le 3/2\right),\\ {}O\left({x}^{9/8+\rho /4}\right)& \left(1/2<\rho \le 1\right),\\ {}O\left({x}^{5/4+\varepsilon}\right)& \left(0<\rho \le 1/2\right)\end{array}} \)
and
\( {\Delta}_{1/2}\left(x;{\zeta}_4\right)=\Omega \left(x{\log}^{1/2}x\right). \)
作者简介
O. Fomenko
St. Petersburg Department of the Steklov Mathematical Institute
Email: Jade.Santos@springer.com
俄罗斯联邦, St. Petersburg