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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 if and only if there is an ε > 0 such that for all k > 1,

$\frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(k)}{k^{1+\varepsilon}}$

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.

[edit] 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:

$\sigma(n)<\exp(\gamma) \cdot n \log\log n$

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.

[edit] References

^ 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.
^ J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
^ Warren D. Smith, A "good" problem equivalent to the Riemann hypothesis, 2005

[edit] External links

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[0] 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.

[1] J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.

[2] Warren D. Smith, A "good" problem equivalent to the Riemann hypothesis, 2005

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