[MO] Residue at ∞ and ∑(-1)^n x^(2^(2^n))

[tommy1729]

...

Also, Caleb. Go easy on me. I'm just spit balling here, and trying to give my two cents. I apologize if I'm being stupid, lmao. But if you go backwards from here; you should be able to find a function \(q(z)\) such that:

\[
\int_{|z| = 1} f(z)\,dz = \int_{|z| = r} f(z) q(z)\,dz\\
\]

And it could be more manageable. I apologize I can't be of more help! These are things I would exclusively wrestle with using the Mellin transform; and what you're detailing relates to an arc \(\gamma\) such that \(\gamma(0) = 0\) and \(\gamma(\infty) = \infty\). And similarly \(\gamma*(0) =0\) and \(\gamma*(\infty) = \infty\). Then if a function is differintegrable we have:

\[
\int_\gamma \vartheta(w)\,dw = \int_{\gamma^*}\vartheta (w)\,dw
\]

Which is just a fancy cauchy's integral theorem. At infinity; if this isn't differintegrable, I conjectured that there was an equivalence:

\[
\lim_{R\to \infty} \int_{\gamma_R}\vartheta(w)\,dw - \int_{\gamma^*_R}\vartheta (w)\,dw = 0\\
\]

Then this function was differintegrable... Where \(\gamma_R\) is \(|\gamma| \le R\). This largely went nowhere but would mean fantastic and huge things for determining when a function is differintegrable. Which would again; talk about how we handle fourier transforms, and fourier coefficients of your residues. Have you considered:

\[
\int_{|z| = 1} f(z) z^k\,dz\\
\]

Which are fourier coefficients at their core.

Either way, don't shoot me. I'm mostly just casually spitballing here. I love your work and I just like to talk.

Regards, James

First, I want to say I believe in that conjecture.
I think it can be proven for smooth curves.
Fractal curves might be the tricky part.

In fact I bet someone already has, then again I did assume in the past alot has been done that turned out to be surprisingly never formally considered.

Im talking about constructible curves here, so do not shoot me with nonconstructible exotic set theoretical curves that require AC.



**Ok BELOW I will talk about poles and residues now , but it is actually more general like log singarities and essential singularities etc **

But anyways, what I really wanted to say is that we are considering residues at the BOUNDARY.

That is another matter than residues on the interior.

The boundary does not take the full range or domain.
For instance the domain may contain + oo , but not - oo.
When mapping infinity to 0 , we get that the point zero is not completely considered but just the positive or negative direction (corresponding to +oo or -oo).

So most theorems in complex analysis are for connected open sets or interiors of connected open sets on the riemann sphere.

But that is not so much the case here.

The angle that cuts the point of the domain is crucial.
If that is even well-defined ( fractal , nonanalytic curves , curves based on AC etc )

Furthermore, one can not simply say f(z) is analytic at the reals and for Im(z) > 0 ** but not analytic for Im(z) < 0 ** and then take contour integral or path integrals on the real line and pretend everything is fine.
Again the point 0 is only partially in the upper plane and the same for its neighbourhood , again a case of edge or boundary.


Not only are you then working on the boundary, what may be an obstacle to the conditions for the theorems ,
but saying that the boundary of analyticity is analytic itself with a straight poker face is dubious magic math.

That is like saying the prime zeta function is analytic at its boundary of analyticity.
And then integrate around some of the poles of the prime zeta near the boundary , even though they are only partially used as poles + there is a dense set of poles around all poles , and pretend the theorems for contour integration around poles are valid.  
And ignore that we did not take the whole neighbourhood of those points.
And then conclude the Riemann Hypothesis is true.

I actually have seen similar "proofs" like that for the prime twin conjecture.
So a big warning here.

( remember in the other thread were you thought every integral around the origin with path a circle at origin with any radius ( even the one corresponding to the natural boundary ) was 0 as extension to cauchy ? (it was false) This seems a similar problem )

 

Dont get me wrong you had some good ideas and the Riemann mapping is a good idea as we mentioned before.

But the R mapping is more of a " restatement " then a solution.
Comparable to the formulas of cauchy linking zero's and integrals of a function , they give alot of insight and be useful , but actually solving NEITHER of them in general.
( in particular  the generalized argument principle in mind )


Look at Calebs pictures again that will probably help.

Not trying to be rude or belittle but the problem is complex.


Let me restate a crucial part of the Argument principle as written in most books  :

... Specifically, if f(z) is a meromorphic function inside and on some closed contour C, and f has no zeros or poles on C, then ...

The problem is NOT magically solved by making a riemann mapping and fourier series and assuming the condition has been altered.

Even stronger the condition is NESSESSARY and is not a weakness of proving power ; counterexamples exist that defy the relaxation of conditions.




regards

tommy1729