Prime numbers with a certain extremal type property

TL;DR

This paper explores the geometric properties of the prime counting function's convex hull, introduces questions based on numerical data, and proves a conditional limit result assuming the Riemann hypothesis.

Contribution

It presents new observations about the sequence of vertices of the convex hull of the prime counting function and proves a limit relation under the Riemann hypothesis.

Findings

The vertices form an infinite sequence with specific properties.

If the Riemann hypothesis holds, then the ratio of consecutive vertices tends to 1.

Numerical data suggests several open questions about the sequence.

Abstract

The convex hull of the subgraph of the prime counting function $x\rightarrow \pi(x)$ is a convex set, bounded from above by a graph of some piecewise affine function $x\rightarrow \epsilon(x)$. The vertices of this function form an infinite sequence of points $(e_k,\pi(e_k))_1^{\infty}$. In this paper we present some trivial observation about the sequence $(e_k)_1^{\infty}$ and we formulate a number of questions resulting from the numerical data. Besides we prove one less trivial result: if the Riemann hypothesis is true, then $\lim\frac{e_{k+1}}{e_k}=1$.