Unitary Multiperfect Numbers in Certain Quadratic Rings

A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\displaystyle{\frac{n}{c}}$. For any integer $k$, the function $\sigma_k^*$ is a multiplicative arithmetic function defined so that $\sigma_k^*(n)$ is the sum of the $k^{th}$ powers of the unitary divisors of $n$. We provide analogues of the functions $\sigma_k^*$ in imaginary quadratic rings that are unique factorization domains. We then explore properties of what we call $n$-powerfully unitarily $t$-perfect numbers, analogues of the unitary multiperfect numbers that have been defined and studied in the integers. We end with a list of several opportunities for further research.