Dirichlet divisor problem on Gaussian integers

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.