## Friday, January 17, 2014

### More On the Dastardly Sum of All Positive Integers

Numberphile's recent video showing that, in some nonridiculous sense,

1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12

is free-falling through the Internet, with all the usual obsession, commotion and insults that usually entails. It won't be long now until someone attributes the proof to the Nazis.

A real mathematican gets involved on Quora, taking, as usual, all the fun out of everything. Better is John Baez's lecture on the number 24, which, being 2 × 12, you know is going to get hinky somewhere -- John, whose footprint in the set {mathematics + physics} is probably as big as anyone's, says the sum "can be made rigorous." He includes this little ditty from the mathematician Niels Henrik Abel, another one of those mathematicians who died way too young (26):

The divergent series are
the invention of the devil,
and it is a shame to base on them
any demonstration whatsoever.

(Abel died after, having contracting tuberculosis in Paris, he took a Christmas trip by sled (!) from Paris to Norway to visit his fiance.) He still has a huge number of things named after him, which leads to the only math joke (of two jokes in total) I can remember:

Q: What's purple and commutes?
A: An Abelian grape.

I still maintain that, because the calculation of the Casimir force involves the number ζ(-3), and because you can relate ζ(s) and ζ(1-s) for s=4, and because experiments verify the Casimir force prediction, the sum of all positive integers is, in some important and real sense, equal to -1/12, at least after all the infinities cancel out.