User:JLAF

This page is just a sandbox really...

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 &gt; 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.