Lines in the prime number graph

TL;DR

This paper investigates the geometric properties of the prime number graph, establishing bounds on the number of line segments needed to cover points and the maximum points on a single line, with results depending on the Riemann Hypothesis.

Contribution

It provides new bounds on the covering line segments and collinear points in the prime number graph, improving previous conjectures and conditional results under RH.

Findings

L(n)=O(n  log log n / \u0000 log n)

B(n) c log n for large n

Under RH, B(n)=O(n^{3/4}(\u0000 log n)^{1/2}) and L(n) c' n^{1/4} ( log n)^{-1/2}

Abstract

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$