@KConrad And the collective concept sense of the term kind of circles back (pun intended) to the O shape, since apparently the German "Ring" ("HRing") was first used as a kind of synonym for "Kreis"

Interesting that Bourbaki ended up using the first letter of the german words for the definitive number systems symbols, considering the political situation in those years, probably Weil was the main responsible for this

I should add that for me the etymology of ring is the one that have puzzled me the most since I was an undergrad, and I must confess that I forced myself to believe that it was just a "cool" word that was arbitrarily chosen by mathematicians to refer to an important mathematical structure, but that really didn't give any information about those structures. I even felt kind of embarassed to ask my professors about its etymology for fear of being told that the reason for the name was too obvious. I feel now a great relief finding out that is a complex etymology after all! thank you!

@KConrad reading from the multiple comments and links connected to this from your link, I think is safe to say that we don't know why Hilbert chose the word Zahlring to denote the set of algebraic integers? I'm leaning towards the hypothesis that the symbol Dedekind used for his equivalent Ordnungs, does look like a "ring" of sorts, than to the competing hypothesis that refers to how expressions of powers of algebraic integers kind of circle back to a member of the ring.

Not what you asked for, but books in a similar category to Oddifreddi's: "Mathematics Unlimited — 2001" and "Beyond & Mathematics: Frontiers and Perspectives"

I wonder why Poincaré's work on Automorphic Functions is not developed in Roy's book

I think the confusion with the question is that Riemann extended the domain of the zeta function from the reals to the complex, but also extended the domain from the complex with real part > 1 to the rest of the complex plane minus {1}

I interpret the question as not about extending functions from the reals to the complex but as to: who was the first person to extend the domain of a complex valued function exactly in those cases where the formula for the original domain doesn't converges or doesn't makes sense in the extended domain, and as such, that person was the first to discover the marvelous fact that there is another formula that makes sense in the extended domain and that preserves continuity with original formula at the boundary of the original domain, as in the case of the riemann zeta function.

@DonuArapura are the abelian functions the only meromorphic functions that can be defined on an abelian variety ?