### Abstract:

This study is placed in the framework of Internal Set Theory. In Chapter 1, we will give a necessary condition for which are simultaneously approximable in the infinitesimal sense. The converse of this condition is also discussed. Let k be a positive integer, and let $W_{k}$ denote the set of all positive integers $n$ such that the number of distinct prime factors of $n$ is greater or equal to $k$. In Chapter 2, we prove the infinity of certain inequalities and equations on the set $W_{k}$ by using $\psi$, $\sigma$, $\tau$ and related functions. In the framework of IST, our working set is an infinite external subset of positive integers. In Chapter 3, we will determine an arithmetic function f for which $f^{N}(n)-\alpha\,f^{N}(n+\ell)$ has infinitely many sign changes on a proper infinite subset of $W_{k}$, where $\alpha$, $N$ and $\ell$ are parameters. This result will be realized by using Dirichlet's Theorem about primes in an arithmetic progression. In Chapter 4, some open problems are posed for further research.