The asymptotic properties of $\varphi (n)$ and a problem related to visibility of Lattice points

Abstract

We look at the average sum of the Euler's phi function $\varphi (n)$ and it's relation with the visibility of a point from the origin.We show that $\forall k\ge 1,k\in \mathbb{N},\exists$ a $k$$\times$$k$ grid in the 2D space such that no point inside it is visible from the origin.We define visibility of a lattice point from a set and try to find a bound for the cardinality of the smallest set S such that for a given $n$ $\in \mathbb{N}$,all points from the $n$$\times$$n$ grid are visible from S.