The Ramanujan Journal

, Volume 29, Issue 1, pp 359–384

On SA, CA, and GA numbers

Authors

  • Geoffrey Caveney
  • Jean-Louis Nicolas
    • Université de Lyon; CNRS; Université Lyon 1; Institut Camille Jordan, Mathématiques
Article

DOI: 10.1007/s11139-012-9371-0

Cite this article as:
Caveney, G., Nicolas, J. & Sondow, J. Ramanujan J (2012) 29: 359. doi:10.1007/s11139-012-9371-0

Abstract

Gronwall’s function G is defined for n>1 by \(G(n)=\frac{\sigma(n)}{n \log\log n}\) where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for all prime factors p of N. We say that N is a GA2 number if G(N)≥G(aN) for all multiples aN of N. In (Caveney et al. Integers 11:A33, 2011), we used Robin’s and Gronwall’s theorems on G to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers ≤5040, and prove that a GA2 number N>5040 exists if and only if RH is false, in which case N is even and >108576.

Keywords

AlgorithmColossally abundantGronwall’s theoremPrime factorRiemann HypothesisRobin’s inequalitySum-of-divisors functionSuperabundant

Mathematics Subject Classification (2000)

11M2611A4111Y55

Copyright information

© Springer Science+Business Media, LLC 2012