In what topological space does the sum of the natural numbers equal minus one twelfth?

It's fashionable for mathematicians to hate on the trope from popular culture, that the sum of the natural numbers equals minus one twelfth.

Their reasons are varied but typically go that $$-\frac1{12}$$ is just the value arrived at by analytic continuation, and popular culture, by abstracting away that fact, arrives at a misleading novelty.

Nevertheless, the popular culturists are in the ascendancy over the pedants at least to the extent that, unless they stipulate the topology of the discrete number line as a subset of the real line, and leave themselves freedom to choose their topology, they are not strictly wrong, provided we forgive them a little ambiguity.

There are of course topologies in which infinite sums of natural numbers (which otherwise diverge on the real number line), converge - such as $$1+4+16+64+\ldots\to-\frac13$$ in the 2-adic number system.

Is it wrong of me to think of analytic continuation or the Ramanujan summation as another example of the same thing - a topology on the complex numbers such that certain sequences converge, and is there a name for, or a definition of, the topology on the natural numbers such that their sum converges to minus one twelfth? If so, is there some algebra over a set containing $$\Bbb N$$ which accommodates this summation, and which is in some sense "complete" in the same way as, or a similar way to, the p-adic numbers?

• I discussed this with a colleague some time ago. He mentioned that Terrence Tao had done significant work on the topic, so that might be a worthwhile avenue to explore. Jun 1 at 17:54
• "Pedants" is a bit harsh. People have been mislead and have asked "Why does $1+2+\cdots=-1/12$?" - or stubbornly cling to the belief that it is true in the usual sense of summation - for a long time. Being pedantic is more than just being correct. The pop-sci expositions of this rarely mention the subtlety, so, essentially just (knowingly?) present incorrect maths to their audience - which is reasonable to hate Jun 1 at 18:07
• In the absence of precise definition one assumes the usual conventions. I've had to explain to multiple of my friends that $1+2+\cdots=-1/12$ is false in the usual sense, and they had no idea that the author could have meant anything else. At least, the author pulls the wool over the audience's eyes (for clicks) if they don't mention any subtlety and present it as ordinary summation Jun 1 at 18:19
• terrytao.wordpress.com/2010/04/10/… Jun 1 at 18:19