Multidimensional exponential divisor function over Gaussian integers

TL;DR

This paper introduces and analyzes generalized multidimensional exponential divisor functions over Gaussian integers, determining their maximal orders and deriving asymptotic formulas for their average behavior.

Contribution

It extends the multidimensional exponential divisor function to Gaussian integers and provides new results on their maximal orders and average asymptotics.

Findings

Determined maximal orders of the generalized functions.

Established asymptotic formulas for average orders.

Extended the classical functions to the Gaussian integer domain.

Abstract

Let $\taue_k \colon \Z\to\Z$ be a multiplicative function such that $ \taue_k(p^a) = \sum_{d_1... d_k=a} 1 $. In the present paper we introduce generalizations of $\taue_k$ over the ring of Gaussian integers $\Zi$. We determine their maximal orders by proving a general result and establish asymptotic formulas for their average orders.