The main purpose of this paper is to define a new additive arithmetic function related to a fixed integer k\geq 1 and to study some of its properties. This function is given by \begin{equation*} \ f_{k}\left( 1\right) =0\text{ and }f_{k}\left( n\right) =\sum_{p^{\alpha}\parallel n}\left( k,\alpha \right) , \end{equation*} such that (a, b) denotes the greatest common divisor of the integers a and b.

The main purpose of this paper is to define a new additive function related to generalized divisors of an integer and to study some of its properties.

In this paper, we obtain asymptotic formula on the "hyperbolic" summation \begin{equation*} \underset{mn\leq x}{\sum }D_{k}\left( \gcd \left( m,n\right) \right) \text{ \ \ }\left( k\in \mathbb{Z}_{\geq 2}\right), \end{equation*} such that $D_{k}\left( n\right) = \dfrac{\tau _{k}\left( n\right) }{\tau_{k}^{\ast }\left( n\right) }$, where $\tau _{k}\left( n\right) =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1$ denotes the Piltz divisor function, and $\tau _{k}^{\ast }\left( n\right) $ is the unitary analogue function of $\tau _{k}\left( n\right) $.

The gcd-sum function is one of the most important functions that has been studied by many researchers in recent years (Broughan, Bordellès, etc.). The gcd-sum function appears in a specific lattice point problem, where it can be used to estimate the number of integer coordinate points under the square root curve. In this paper, we give an average order of the gcd-sum function over the set of squares.