extraordinary number

Define the function G for integers n>1 by


where σ(n) is the sum of the positive divisorsMathworldPlanetmathPlanetmath of n.  A positive integer N is said to be an extraordinary number if it is composite and


for any prime factorMathworldPlanetmath p of N and any multiple aN of N.

It has been proved in [1] that the Riemann HypothesisMathworldPlanetmath is true iff 4 is the only extraordinary number.  The proof is based on Gronwall’s theorem and Robin’s theorem.


  • 1 Geoffrey Caveney, Jean-Louis Nicolas, Jonathan Sondow:  Robin’s theorem, primes, and a new elementary reformulation of the Riemann Hypothesis.  - Integers 11 (2011) article A33;  available directly at http://arxiv.org/pdf/1110.5078.pdfarXiv.
Title extraordinary number
Canonical name ExtraordinaryNumber
Date of creation 2013-03-22 19:33:41
Last modified on 2013-03-22 19:33:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 11M26
Classification msc 11A25
Related topic PropertiesOfXiFunction