Dirichlet divisor problem on Gaussian integers

Abstract

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.

10 pages