Prime numbers with a certain extremal type property

Abstract

The convex hull of the subgraph of the prime counting function $x\to \pi (x)$ is a convex set, bounded from above by a graph of some piecewise affine function $x\to \epsilon (x)$. 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 $\lim \frac{e_{k+1}}{e_k}=1$.