On SA, CA, and GA numbers

Geoffrey Caveney (1) , Jean-Louis Nicolas (2) , Jonathan Sondow (2)

1 Chercheur indépendant
2 ICJ - Institut Camille Jordan

Gronwall's function $G$ is defined for $n>1$ by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where $\sigma(n)$ is the sum of the divisors of $n$. We call an integer $N>1$ a~\emph{GA1 number} if $N$~is composite and $G(N) \ge G(N/p)$ for all prime factors~$p$ of~$N$. We say that $N$ is a \emph{GA2 number} if $G(N) \ge G(aN)$ for all multiples $aN$ of~$N$. In \cite{CNS}, 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 $\le5040$, and prove that a GA2 number $N>5040$ exists if and only if RH is false, in which case $N$ is even and $>10^{8576}$.

Mots clés: Algorithm; Colossally abundant; Gronwall's theorem; Prime factor *Riemann Hypothesis; Robin's inequality; Sum-of-divisors function *Superabundant

Domaines: Théorie des nombres [math.NT]

Geoffrey Caveney, Jean-Louis Nicolas, Jonathan Sondow. On SA, CA, and GA numbers. Ramanujan Journal, 2012, 29, pp.359-384. ⟨10.1007/s11139-012-9371-0⟩. ⟨hal-00920484⟩