Superabundant Numbers and the Riemann Hypothesis

For a lively exposition of this theorem and its connection to the Riemann Hypothesis see [5]. In this note, we propose a method that will establish explicit upper bounds for σ(n)/en log log n. Our main observation is that the least number violating the inequality (2) should be a superabundant number. A positive integer n is said to be superabundant if σ(m)/m < σ(n)/n for all m < n. The first 20 superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080. The sequence of superabundant numbers is the sequence A004394 in Sloane’s Encyclopedia [8]. The list of the first 500 superabundant numbers is available at [6], where