Table of Contents
Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 491627, 11 pages
http://dx.doi.org/10.1155/2013/491627
Research Article

Efficient Prime Counting and the Chebyshev Primes

1Institut FEMTO-ST, CNRS, 32 Avenue de l’Observatoire, F-25044 Besançon, France
2Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
3Mathematical Department, King Abdulaziz University, Jeddah, Saudi Arabia

Received 17 October 2012; Revised 28 January 2013; Accepted 29 January 2013

Academic Editor: Pantelimon Stǎnicǎ

Copyright © 2013 Michel Planat and Patrick Solé. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The function where is the logarithm integral and the number of primes up to is well known to be positive up to the (very large) Skewes' number. Likewise, according to Robin's work, the functions and , where and are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH) holds. One introduces the jump function at primes and one investigates , , and . In particular, , and for . Besides, for any odd , an infinite set of the so-called Chebyshev primes. In the context of RH, we introduce the so-called Riemann primes as champions of the function (or of the function ). Finally, we find a good prime counting function , that is found to be much better than the standard Riemann prime counting function.