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

[Caleb]

Do you (implicitly) claim to have reduced the problem to fractional differentiation ?

I think you forgot a summation symbol around the third equation.



Am I correct in thinking if there is only 1 pole and it is at the boundary then only the angle of the curve at that point matters ?
Now wait, make that the 2 angles made by the corner , that touches the poles ; the 2 angles are with respect to the cartesian axes.

Also what about singularities and branch cuts at the boundary ?
Is this really only about poles ?


You often talk about doing analytic number theory, but how does this relate ?

Sorry if I am a bit slow.


regards

tommy1729

The connection between fractional integration is the following. We have that 
\[f(x) = \sum_{n=0}^\infty a_n x^n\]
Where \( a_n = \frac{f^{(n)}(0)}{n!} \). My work has reduced the evaluation of certain integrals into what is essentially the evaluation of \( a_\frac{1}{2} \). Some earlier conversations I've had with James suggests to me that the DifferIntegral is the uniquely correct way to interpolate the sequence \( a_n \). Therefore, evaluation of \( a_\frac{1}{2} \) should be equal to \( \frac{f^{(\frac{1}{2})}}{(\frac{1}{2})!} \), which is the half derivative.  

Also, you are right about the missing summation sign-- I'll try to fix everything later today, and redo my explanation a bit, and maybe try to do some computations to see if the math checks out in other cases. 

The post is actually only about essential singularities. It won't work for poles, or branch cuts. Actually, for poles, the natural extension of this idea gives something rather trivial. If there is a pole after doing the \( dz \to d \frac{1}{z} \) map, then that means you must have started with a polynomial. In which case, this method gives exactly what you would expect. For odd powers of \(k\), it evaluates to zero (which is what you would expect, since, morally, \( \int_{-\infty}^\infty x^k dx = 0 \). For functions that include an even power, the integral evaluates to infinity.

James has a better grasp of how this connects to analytic number theory (he mentioned a bit on this idea when he talked about the prime number theorem in his last post). For now, I don't have a good enough idea on how these ideas work out to actually claim that this method has applications for anything whatsoever-- it might turn out to be completely useless. 

Actually, one case I'm interested in trying to apply this is to compute the integral 
\[ \int_{-\infty}^\infty (\eta(x^2) - 1) dx \]
Where eta is the Dirichlet Eta function (which is the alternating version of the zeta function). Presumably, the answer to this integral should be given in terms of the half integral of the Eta function, which seems interesting to me (if its true that is).