Prime numbers with a certain extremal type property

Edward Tutaj

Abstract


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.

Keywords


prime numbers; prime counting function; Riemann hypothesis

Mathematics Subject Classification (2010)


11N05

References


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