高阶超丰数
Higher-order colossally abundant numbers
AI总结：
本文提出高阶超丰数，通过几何坐标定义嵌套子集，研究其性质及与黎曼猜想和罗宾不等式的关联，给出增长率和整除性结果。
AI中文摘要：
超丰数相对于其大小具有较大的除数之和。它们还允许使用平面凸包的支撑线进行几何描述。从这一描述出发，我们改变坐标以获得这些数的嵌套子集，称之为高阶超丰数。其定义不依赖于黎曼猜想。我们给出了在任一这些子集上罗宾不等式成立当且仅当黎曼猜想为真的条件。某些所得族在每个层级都有无限集但交集为空。对于两类坐标，我们确定了最小成员增长的速度。我们还证明了每个固定的正整数整除所有足够高阶的成员。在凹性假设下，最小成员构成整除链；对于幂次横坐标，相继商具有无界个数的素因子，并具有显式的上极限增长率。在第二类族中，归一化除数和的每个全局最大值都被保留，且最小成员趋于无穷当且仅当黎曼猜想为真。我们给出数值例子，并解释拉马努金-尼古拉斯界如何产生终止链。
英文摘要：
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.
