Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 750–757

# Lattice Points in the Four-Dimensional Ball

• O. M. Fomenko
Article
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
$${\Delta}_{1/2}\left(x;{\zeta}_4\right)=\Omega \left(x{\log}^{1/2}x\right).$$

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