Prime numbers with a certain extremal type property

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 $\lim\frac{e_{k+1}}{e_k}=1$.