On Riesz means of the coefficients of Epstein's zeta functions

Let $r_k(n)$ denote the number of lattice points on a $k$-dimensional sphere of radius $\sqrt n$.The generating function
$$ \zeta_k(s)=\sum^\infty_{n=1}r_k(n)n^{-s},\ k\geq2, $$
is Epstein's zeta-function. Let $k=3$. We consider the Riesz mean of the type
$$ D_\rho(x;\zeta_3)=\frac1{\Gamma(\rho+1)}\sum_{n\leq x}(x-n)^\rho r_3(n) $$
for any fixed $\rho>0$ and define the error term $\Delta_\rho(x;\zeta_3)$ by
$$ D_\rho(x;\zeta_3)=\frac{\pi^{3/2}x^{\rho+3/2}}{\Gamma(\rho+5/2)}+\frac{x^\rho}{\Gamma(\rho+1)}\zeta_3(0)+\Delta_\rho(x;\zeta_3). $$
A result of K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) gives
$$ \Delta_\rho(x;\zeta_3)= \begin{cases} O(x^{1/2+\rho/2)}&(\rho>1),\\ \Omega_\pm(x^{1/2+\rho/2})&(\rho\geq0). \end{cases} $$
In § 2 one proves that
$$ \Delta_\rho(x;\zeta_3)= \begin{cases} O(x\log x)&(\rho=1),\\ O(x^{2/3+\rho/3+\epsilon})&(1/2<\rho<1),\\ O(x^{3/4+\rho/4+\epsilon})&(0<\rho\leq1/2). \end{cases} $$
In § 3 one mentions a few examples for which results of § 2 are applicable.
In § 4 one investigates Riesz means of the coefficients of $\zeta_k(s)$, $k\geq4$.