You seem to have put quite an extensive amount of biographical research into this. I have never seen these photos in this high quality on the internet before.
As a maths person and a bit of a Grothendieck fan, thanks!
"New mathematics" is a bit of an overstatement. There is a lot more to maths than continued fractions.
It bugs me a bit that the authors write (I would love for anyone affiliated with the project to talk to me about this) "Any new conjecture, proof, or algorithm suggested will be named after you.". No offense, but there are very few mathematicians out there with that kind of a world view.
Seems a bit like a high school project without proper guidance from a mathematician.
It's a theorem of analytic number theory. It's just a special value of the ζ-function. People misunderstand how analytic continuation works and therefore (wrongly) interpret ζ(-1) as the divergent series 1+2+3+... (The correct interpretation is as the unique analytic continuation of the holomorphic function $\sum_{n\geq 1}\frac{1}{n^s}$ defined on the half-plane $\Re(s)>1$ to $\mathbb{C}\setminus\{1\}$.)
It's actually a pretty simple consequence of the functional equation for ζ and a few special values of Γ. That in turn comes from a theta-function identity which can be proven using Poisson's summation formula.
What I'm trying to say is that it's legit maths, that has been distorted due to the shock value of writing the equation "1+2+3+... = -1/12".
If you're looking for a reference, go to Davenport's "Multiplicative Number Theory". It's short, self-contained, and extremely well-written. Serre's "A Course in Arithmetic" should also work.
As a maths person and a bit of a Grothendieck fan, thanks!