colossally abundant number

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

where σ denotes the divisor function. The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, ... (sequence A004490 in OEIS); all colossally abundant numbers are also superabundant numbers, but the converse is not 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 equivalent to the assertion that the following inequality is true for n > 5040:

where γ is the Euler–Mascheroni constant.

This result is due to Robin^[1].

Lagarias^[2] and Smith^[3] discuss this and similar formulations of the RH.

References

1. ^ G. Robin, "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées 63 (1984), pp. 187-213.
2. ^ J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
3. ^ Warren D. Smith, A "good" problem equivalent to the Riemann hypothesis, 2005

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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