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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).
The Limit Superior, or "lim sup", is best explained by the picture to the right of this text.
The divisor function , σ しぐま (n ), is defined as the sum of the positive divisors of n , or
σ しぐま
(
n
)
=
∑
d
|
n
d
.
{\displaystyle \sigma (n)=\sum _{d|n}d\,\!.}
The asymptotic growth rate of the divisor function can be expressed by:
lim sup
n
→
∞
σ しぐま
(
n
)
n
log
log
n
=
e
γ がんま
,
{\displaystyle \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)[ edit ]
A number n is colossally abundant if and only if there is an ε いぷしろん > 0 such that for all k > 1,
σ しぐま
(
n
)
n
1
+
ε いぷしろん
≥
σ しぐま
(
k
)
k
1
+
ε いぷしろん
{\displaystyle {\frac {\sigma (n)}{n^{1+\varepsilon }}}\geq {\frac {\sigma (k)}{k^{1+\varepsilon }}}}
where σ しぐま denotes the divisor function .
There are infinitely many Colossally Abundant Numbers.