Prime numbers with a certain extremal type property

• Main Author: Edward Tutaj
• Format: Article
• Language: deu
• Published: Wydawnictwo Naukowe Uniwersytetu Pedagogicznego 2019-02-01
• Series: Annales Universitatis Paedagogicae Cracoviensis: Studia Mathematica
• Subjects: prime numbers; prime counting function; Riemann hypothesis.
• Online Access: http://studmath.up.krakow.pl/index.php/studmath/article/view/307

The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → ε(x). The vertices of this function form an infinite sequence of points (ek,π(ek))1∞. The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two - it seems - interesting results. First states that if the Riemann Hypothesis is true, then lim(ek+1/ek)=1. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.