On SA, CA, and GA numbers

TL;DR

This paper investigates special classes of integers called GA1 and GA2 numbers, exploring their properties, algorithms for computation, and their connection to the Riemann Hypothesis, extending previous work on Gronwall's function and related number classes.

Contribution

It provides new algorithms for identifying GA1 numbers, characterizes GA2 numbers, and establishes their relationship with the Riemann Hypothesis, expanding the understanding of these special number classes.

Findings

Smallest GA1 with >2 prime factors is 183783600

Smallest odd GA1 is 1058462574572984015114271643676625

Existence of large GA2 numbers beyond 5040 is equivalent to RH being false

Abstract

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 arXiv 1110.5078, 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. Here, 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 $\le 5040$, and prove that a GA2 number $N>5040$ exists if and only if RH is false, in which case $N$ is even and $>10^{8576}$.