I referred to this thread in
http://math.eretrandre.org/tetrationforu...43#pid6943
For simplicity lets replace arc2sinh(x/2) with ln(x).
This is a bit informal handwaving, but for the time being it might be good enough.
Let f(x) = ln(x) ^ ln(ln(x)) ^ ln^[3](x) ^ ...
As often in math we get the weird result that something is true IFF something else is true.
Keeping this in mind :
This might be handwaving but lets go :
Let eps > 0
( the famous epsilon )
Assume for sufficiently large x :
x^eps < f(x) < x^A
for some real A > eps.
(the epsilon assumption)
This is more powerfull than you might expect ! :
We know ln(x^b(x)) / ln(x) = b(x)
Based on that :
f(x) = ln(x) ^ ln(ln(x)) ^ ln^[3](x) ^ ...
ln(f(x)) / ln(x)
= ln( ln(x) ^ ln(ln(x)) ^ ln^[3](x) ^ ... ) / ln(x)
= (ln(ln(x)) * ln(ln(x)) ^ ln^[3](x) ^ ...) / ln(x)
use f(x) = x^b(x)
= ln(ln(x)) * ln(x)^b(ln(x)) * (ln(x))^-1
=> the best constant fit for b(x) = b(ln(x)) = b => ( existance of b follows from the epsilon assumption )
=> b = 1 because ln(ln(x)) * ln(x) / ln(x) = ln(ln(x))
and ln(ln(x)) is smaller than a power of ln(x).
By a similar logic we can improve : f(x) = x^b(x) = x^b / ln(x)
This gives us :
= ln(ln(x)) * ln(x)^b(ln(x)) * (ln(x))^-1
= ln(ln(x)) * f(ln(x)) * (ln(x))^-1
= ln(ln(x)) * ln(x)^b / ln(ln(x)) * (ln(x))^-1
= 1.
= b.
QED.
Hence if x^eps < f(x) < x^A
then f(x) = x / (ln(x))^(1+o(1)).
WOW.
THIS IS THE PNT FOR POWER TOWERS !!
However like I said this is a bit handwaving.
Because ... " replace arc2sinh(x/2) with ln(x) " ?
Is x^eps < f(x) < x^A TRUE ???
Funny thing is also that currently the mellin transform is considered for tetration.
The same mellin transform used in the proof for PNT !
So maybe a mellin transform can be used for this power tower problem as well ???
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Remark :
Its not even clear f(x) is analytic ?
I considered Taylor series , four series and dirichlet.
And taking derivatives.
But if f(x) is not analytic ...
That might also be problematic for the mellin transform.
On the other hand f(x) is C^oo.
So I believe taking derivatives is justified.
This brings me to another idea :taking the logarithmic derivative of f(x).
That might help decide if the epsilon assumption is true !!
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regards
tommy1729