Colossally abundant number

Wikipedia, the free encyclopedia - Cite This Source

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,

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

where σ denotes the divisor function. The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, ... ; 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 $gamma$ is the Euler–Mascheroni constant.

This result is due to Robin.

Lagarias and Smith discuss this and similar formulations of the RH.

References

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

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)
This article is licensed under the GNU Free Documentation License.
Last updated on Wednesday March 05, 2008 at 04:38:08 PST (GMT -0800)
View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation