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


Goldfeld, Dorian. "The elementary proof of the prime number theorem: an historical perspective." In Number theory (New York, 2003), 179-192. New York: Springer, 2004.

McNew, Nathan. "The convex hull of the prime number graph." In Irregularities in the distribution of prime numbers, 125-141. Cham: Springer, 2018.

McNew, Nathan. Multiplicative problems in combinatorial number theory. Ph.D. diss., Dartmouth College, 2015.

Odlyzko, Andrew, and Michael Rubinstein, and Marek Wolf. "Jumping champions." Experiment. Math. 8, no. 2 (1999): 107-118.

Pomerance, Carl. "The prime number graph." Math. Comp. 33, no. 145 (1979): 399-408.

Montgomery, Hugh L., and Stan Wagon. "A heuristic for the prime number theorem." Math. Intelligencer 28, no. 3 (2006): 6–9.

Tutaj, Edward. "Prime numbers with a certain extremal type property." arXiv (2014): arXiv:1408.3609 .

Tutaj Edward. "Pewna uwaga o wykresie funkcji zliczaj¸acej liczby pierwsze." Tarnow Scientific Colloquia (2018): 55-77.

Zhang, Yitang. "Bounded gaps between primes." Ann. of Math. (2) 179, no. 3 (2014): 1121-1174.


Full Text: PDF

Download statistics: 865

Licencja Creative Commons
This article by Edward Tutaj is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej