A von Staudt-type formula for ${\displaystyle \sum _{z\in {\mathbb{Z}}_n[i]}{z}^k}$

In this paper we study the sum of powers in the Gaussian integers ${\mathbf{G}}_k(n):={\sum }_{a,b\in [1,n]}(a+bi{)}^k$. We give an explicit formula for ${\mathbf{G}}_k(n)(\mathrm{m}\mathrm{o}\mathrm{d}n)$ in terms of the prime numbers $p \equiv 3 \pmod 4$with$p \mid \mid n$and$p-1 \mid k$, similar to the well known one due to von Staudt for ${\sum }_{i=1}^n{i}^k(\mathrm{m}\mathrm{o}\mathrm{d}n)$. We apply this formula to study the set of integers $n$ which divide ${\mathbf{G}}_n(n)$ and compute its asymptotic density with six exact digits: $0.971000\dots$.