Lattice points in the four-dimensional ball

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 236-246

Résumé

Let $r_4(n)$ denote the number of representations of $n$ as a sum of $4$ squares. The generating function $\zeta_4(s)$ is Epstein's zeta function. We consider the Riesz mean $$ D_\rho(x;\zeta_4)=\frac1{\Gamma(\rho+1)}\sum_{n\leq x}(x-n)^\rho r_4(n) $$ for any fixed $\rho>0$ and define the error term $\Delta_4(x;\zeta_4)$ by $$ D_\rho(x;\zeta_4)=\frac{\pi^2x^{2+\rho}}{\Gamma(\rho+3)}+\frac{x^\rho}{\Gamma(\rho+1)}\zeta_4(0)+\Delta_\rho(x;\zeta_4). $$ In § 2 one proves that $$ \Delta_4(x;\zeta_4)= \begin{cases} O(x^{1/2+\rho+\epsilon})(1\rho\leq3/2),\\ O(x^{9/8+\rho/4})(1/2\rho\leq1),\\ O(x^{5/4+\epsilon})(0\rho\leq1/2). \end{cases} $$ In § 3 one proves that $$ \Delta_{1/2}(x;\zeta_4)=\Omega(x\log^{1/2}x). $$