Remark on Gottfried's "problem with an infinite product" power tower variation

[tommy1729]

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