Asymptotics for numbers of line segments and lines in a square grid

TL;DR

This paper derives asymptotic formulas for counting line segments and lines with at least or exactly q points in an n by n grid, including sharper estimates under the Riemann hypothesis.

Contribution

It provides new asymptotic formulas for the number of line segments and lines in a grid, extending known cases and assuming the Riemann hypothesis for sharper results.

Findings

Asymptotic formula for line segments connecting q+1 points

Sharper formula assuming Riemann hypothesis

Generalization of the case q=2 to broader q values

Abstract

We present an asymptotic formula for the number of line segments connecting q+1 points of an nxn square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q=2 is so generalized.