Higher-order colossally abundant numbers

Abstract

Colossally abundant numbers have large sums of divisors relative to their size. They also admit a geometric description using supporting lines of a planar convex hull. Starting from this description, we change coordinates to obtain nested subsets of these numbers, which we call colossally abundant numbers of higher order. Their definition does not depend on the Riemann hypothesis. We give conditions under which Robin's inequality holds on any one of these subsets if and only if the Riemann hypothesis is true. Some of the resulting families have infinite sets at every level but an empty intersection. For two classes of coordinates, we determine how fast the least members grow. We also prove that every fixed positive integer divides all members of sufficiently high order. Under a concavity assumption, the least members form a divisibility chain; for power abscissas, successive quotients have unbounded numbers of prime factors, with an explicit limsup growth rate. In a second class of families, every global maximum of the normalized divisor sum is retained, and the least members tend to infinity if and only if the Riemann hypothesis is true. We give numerical examples and explain how the Ramanujan--Nicolas bounds produce terminating chains.