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

[Caleb]

I AM GOING TO DO SOMETHING I DONT KNOW THE IMPORTANCE OF, JUST FOR CALEB! LISTEN!


We are going to define the Dirichlet \(\eta\) function, as the function:

\[
\eta(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{-s}\\
\]

This function is holomorphic for \(\Re(s) > 0\). Additionally, we know that it is \(O(s)\) bounded for \(\Re(s) > 0\). Which means, there exists a constant \(C_\delta >0\) such that \(|\eta(s)| < C_\delta |s|\) for \(\Re(s) > \delta\) when \(\delta>0\).

Now your first instinct might tell you to "fractionally differentiate" \(\eta\). But that's wrong. We instead take:

\[
\vartheta(w) = \sum_{n=0}^\infty \eta(1+n) \frac{(-w)^n}{n!} = \frac{1}{2\pi i} \int_{1/2-i\infty}^{1/2+i\infty} \Gamma(s) \eta(1-s) w^{-s}\,ds\\
\]

Where:

\[
\frac{d^{s-1}}{dw^{s-1}}\Big{|}_{w=0} \vartheta(w) = \eta(s)\\
\]

---

Here is where advanced number theory pokes its head. I am just trying to describe the problem, I am not trying to solve it. For you to describe singularities within the function \(\eta(\infty)\); within these fucking crazy beautiful residues your looking at.... You're solving very deep problems that I don't think you are aware of; and that's why I'm doubtful of some of the shenanigans you're doing.

BUT BY GOD IF YOU'RE NOT AT LEAST ON TO SOMETHING!

Dirichlet's function \(\eta(s)\) is meromorphic on \(\mathbb{C}\)......... Let's just move some integrals around and pull out some residues         



This is true and unique and Hilbert Special and fucking JON VON NEUMANNN UNIQUE so long as we assume that these functions are "Ramanujan bounded". Or we subject ourselves to advanced fourier theory. CARLSON WON A FIELDS MEDAL FOR FINDING THE FOURIER RIGOR VERSION OF RAMANUJAN'S RAW.

So I Fucking love this Caleb! But just know, the fucking devil's in the details. And you're going to have to go case by case; because this relates to highly advanced analytic number theory.

Fucking awesome, regards, James

EDIT:

Forgot to mention; all it takes to turn your integrals into my integrals is a variable change. I mean,

\[
\begin{align}
\frac{d^s}{dw^s}\big{|}_{w=0}f(w^2) &= \frac{1}{\Gamma(-s)} \int_{\gamma} f(w^2) w^{-s-1}\,dw\\
&=\frac{1}{2\Gamma(-s)} \int_{\gamma} f(u) u^{-(s+1)/2-1/2}\,du\\
&= \frac{\Gamma(-s/2)}{2\Gamma(-s)} \frac{d^{s/2}}{dw^{s/2}}\big{|}_{w=0}f(w)
\end{align}
\]

I'm still just rapidly posting these things but fuck this is beautiful.

A short thought is that we can re-express fractional derivatives in terms of Cauchy's integral formula. We have that 
\[ a_n = \int_C \frac{f(z)}{z^n} dz \]
The tempting thing is to plug in non-integer values of n. I tried this before, and it didn't work. Now that I know more about complex analysis, its clear why this niave approach doesn't work-- there is a branch cut at non-integer values of \( n \). I believe we get the desired behaviour if we take \( C \) be a contour that goes around the branch cut, for instance, a Hankel contour. 

So I think this is one way we might start analyzing these things seriously. We can define fractional derivatives in terms of the fractional Cauchy's integral formula, and test if it satisfies the desired behaviour. I guess this also explains why all the integer derivatives of local, whereas the non-integer ones are non-local-- its because this branch cut means the shape of the contour has to change to something that depends on more than just the local behaviour of the function.

Also, this is far in the future and certainly jumping the gun, but I think this fractional derivative idea might fit really well with some of the lambert series stuff we did before. Whats interesting is that many lambert series have that 'gateway' along the real line. That gateway should capture enough behaviour to be able to take fractional derivatives, so perhaps even though fractinoal derivatives are non-local, it seems we can still compute them in a  reasonable way beyond natural boundaries. I'm just spitballing here, it requires much more careful thought to determine if these thigns are actually meaningful.