Lattice Points in the Four-Dimensional Ball

Resumo

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 Δ_ρ(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

\( {\Delta}_{1/2}\left(x;{\zeta}_4\right)=\Omega \left(x{\log}^{1/2}x\right). \)