Searching for an asymptotic to exp[0.5]

[sheldonison]

As promised in the binairy partition function thread :

http://math.eretrandre.org/tetrationforu...70#pid8070

I Will explain the connection between fake function theory and the binary partition function.

Lets use Jay's function J(x).

It is clear that J should be a good fake for the binary part function.

Also fake(J(x)) should be close to J(x).

Notice J(x) satisfies all neccessary conditions , even those for conjecture B and TPID 17.

All conditions for any method used so far.

J(x) grows much slower then exp but much faster then exp.

The growth of J(x) is 0. ( this is how tetration relates )

J(x) is close to exp( ln^2 (x) ).
But we do not even need this here.

J ' (x) = t(x) = J(x/2).
....

Remember, we're interested in the function g(x) = ln(f(exp(x)). Here, g(x)=ln(J(exp(x))). Then g'(h_n)=n. That's why you're getting nonsense results. If you do it correctly, using the approximation function:

\( f(x)=\exp\( (\ln(x))^2 \) \;\;\; g(x) = \ln (f(\exp(x))) \)

\( g(x)=x^2 \;\;\; g'(x)=2x \;\;\; g''(x)=2 \;\;\; h_n=\frac{n}{2} \)

\( a_n \approx \frac{\exp(g(h_n) - n h_n)}{\sqrt{2 \pi g''(h_n) }} \;\;\; \approx \frac{ \exp(-\frac{n^2}{4}) }{ \sqrt{4 \pi} } \)

Then you'll find the entire "fake function" approximation gives ideal results, using the Gaussian approximation. Of course, the partition function itself is non-analytic, so there's a limit to how well you can approximate it. I haven't done the calculations for the partition function itself in a while, but I did them once; I could post the Taylor series again. I find this particular function with \( g(x)=x^2 \) to be really interesting for fake functions precisely because it has such a nice closed form, and because the Gaussian approximation for this function is exactly correct! It also turns out this function has an infinite converging Laurent series, and an exactly defined error term too! See very closely related post#85 http://math.eretrandre.org/tetrationforu...3&pid=7413