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Background

An illustration of limit superior and limit inferior. The sequence x_n is shown in blue. The two red curves approach the limit superior and limit inferior of x_n, shown as solid red lines to the right. The inferior and superior limits only agree when the sequence is convergent (i.e., when there is a single limit).

Limit Superior

The Limit Superior, or "lim sup", is best explained by the picture to the right of this text.

Divisor Function

The divisor function, σ(n), is defined as the sum of the positive divisors of n, or

$\sigma (n)=\sum _{d|n}d\,\!.$

Grönwall's Theorem

The asymptotic growth rate of the divisor function can be expressed by:

$\limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\,\log \log n}}=e^{\gamma },$

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913.

http://mathworld.wolfram.com/images/eps-gif/GronwallsTheorem_1000.gif

Colossally Abundant Numbers (CAs)

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.

There are infinitely many Colossally Abundant Numbers.