  We improve existing estimates of moments of the Riemann zeta function. As a
consequence, we are able to derive new estimates for the asymptotic behaviour
of $\sum_{N \alpha \le x} \mathfrak{t}_k(\alpha)$, where $N$ stands for the
norm of a complex number and $\mathfrak{t}_k$ is the $k$-dimensional divisor
function on Gaussian integers.
