Colossally abundant number

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In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. Formally, a number n is colossally abundant iff there is an ε > 0 such that for all k > 1,

[\frac}\geq\frac}]

where σ denotes the divisor function. The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, ... (sequence [[OEIS:A004490|A004490]] in OEIS); all colossally abundant number are also superabundant numbers, but the converse is not generally true.

Properties

All colossally abundant numbers are Harshad numbers.

Relation to the Riemann hypothesis

If the Riemann hypothesis is false, a colossally abundant number will be a counterexample. In particular, the RH is equivilent to the assertion that the sigma, the sum of the divisors of n, follows this constraint for n >= 5041:

[\sigma(n)<\exp(\gamma) \cdot n \log\log n]

This result is due to RobinG. Robin, "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées 63 (1984), pp. 187-213.

LagariasJ. C. Lagarias, [An elementary problem equivalent to the Riemann hypothesis], American Mathematical Monthly 109 (2002), pp. 534-543. and SmithWarren D. Smith, [A "good" problem equivalent to the Riemann hypothesis], 2005 discuss this and similar formulations of the RH.

See also

• Superabundant number
• Highly abundant number
• Abundant number
• Deficient number

External links

• [Keith Briggs on colossally abundant numbers and the Riemann hypothesis]
• [MathWorld entry]
• [Notes on the Riemann hypothesis and abundant numbers]
• [More on Robin's formulation of the RH]

 

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