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

TL;DR

This paper investigates the average behavior of Euler's phi function and explores the visibility of lattice points from the origin, establishing bounds and existence results for visibility within grid structures.

Contribution

It introduces new results on the existence of non-visible points in grids and bounds for the minimal set ensuring visibility of all points from a given set.

Findings

Existence of k×k grids with no visible points from the origin for all k ≥ 1

Bounds on the minimal set size for visibility of all points in an n×n grid

Analysis of the average sum of Euler's phi function in relation to lattice point visibility

Abstract

We look at the average sum of the Euler's phi function $\phi{(n)}$ and it's relation with the visibility of a point from the origin.We show that $\forall{\hspace{0.05in}{k} \ge{1}},k\in\mathbb{N},\exists$ 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.