Forty Minutes with a Fields Medallist
t5eiitm.org
t5eiitm.org
> After the talk about “Poetry, Drumming and Mathematics”, we arrived at the appointed time at the Bose–Einstein Guest House to find our man, with his mother, standing transfixed by the deer grazing on the lawns. We managed to prise him away for a while — here’s what followed.
is meant to provide that intro/context; but, like you and the grandparent, I was puzzled about who 'our man' was.
For those who wonder like I did why the probability of picking a square-free number is 6/Pi^2, it's explained on wikipedia: https://en.wikipedia.org/wiki/Square-free_integer#Distributi...
http://ocw.mit.edu/courses/mathematics/18-786-topics-in-alge...
If you can explain, convincingly and geometrically, why pi is present in this formula then I would be VERY VERY interested to read.
"Euler's original derivation of the value π2/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series."
Sorry, not very convincing or geometric, and I'm sure someone else can provide a better answer, but that's how I visualize it.
The sum of the inverse fourth powers is pi^4/90. Sixth powers, pi^6/915. The pattern keeps going for even powers and not for odd.
I don't think there is a geometric proof. But if there were, it would be incredible.
1) You accept Parseval's identity: by changing between orthogonal basis, the energy (sum of absolute squared values) remains the same.
2) You find that the energy of the periodic function x mod pi is sum(1/n^2).
3) Evaluating the integral gives pi^2/6!
The square is neatly explained because it involves energies. I guess this is why the pattern works only for even powers.