<abstract xmlns="http://www.w3.org/1999/xhtml">The convex hull of the subgraph of the prime counting function <italic>x → π</italic>(<italic>x</italic>) is a convex set, bounded from above by a graph of some piecewise affine function <italic>x →</italic> (<italic>x</italic>). The vertices of this function form an infinite sequence of points <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub><mo>,</mo><mi>π</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub><mo>)</mo><mo>)</mo></mrow></mrow><mn>1</mn><mo>∞</mo></msubsup></mrow></math><tex-math>$({e_k},\pi ({e_k}))_1^\infty $</tex-math></alternatives></inline-formula>. The elements of the sequence (ek)1∞ shall be called <italic>the extremal prime numbers</italic>. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that <italic>if the Riemann Hypothesis is true, then</italic><inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub><mo>+</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math><tex-math>${{{e_k} + 1} \over {{e_k}}} = 1$</tex-math></alternatives></inline-formula>. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.</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 → (x). The vertices of this function form an infinite sequence of points (ek,π(ek))1∞$({e_k},\pi ({e_k}))_1^\infty $. The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, thenek+1ek=1${{{e_k} + 1} \over {{e_k}}} = 1$. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

Prime numbers with a certain extremal type property

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Prime numbers with a certain extremal type property"}

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 →...

Prime numbers with a certain extremal type property

Published Online: Feb 23, 2019

Page range: 127 - 151

Received: Sep 12, 2018

Accepted: Jan 13, 2019

DOI: https://doi.org/10.2478/aupcsm-2018-0010

Keywords
prime numbers, prime counting function, Riemann hypothesis

© 2018 Edward Tutaj, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Prime numbers with a certain extremal type property

Published Online: Feb 23, 2019

Page range: 127 - 151

Received: Sep 12, 2018

Accepted: Jan 13, 2019

DOI: https://doi.org/10.2478/aupcsm-2018-0010

Keywordsprime numbers, prime counting function, Riemann hypothesis

© 2018 Edward Tutaj, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
prime numbers, prime counting function, Riemann hypothesis