Dirichlet divisor problem on Gaussian integers

TL;DR

This paper advances the understanding of divisor sums over Gaussian integers by improving estimates related to the Riemann zeta function, leading to new asymptotic results for divisor functions in the complex integer domain.

Contribution

It introduces improved estimates for moments of the Riemann zeta function and applies these to derive new asymptotic formulas for divisor sums over Gaussian integers.

Findings

Enhanced estimates for moments of the Riemann zeta function.

New asymptotic behavior results for divisor sums in Gaussian integers.

Improved bounds for the Dirichlet divisor problem in the Gaussian integer setting.

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.