Add up the reciprocals of every perfect cube -- 1 + 1/8 + 1/27 + 1/64 + … -- and the sum doesn't blow up. It converges to 1.2020569…, a number called Apéry's constant. And after centuries of work, mathematicians still cannot tell you what it actually is.
TIMESTAMPS:
0:00 Intro — summing reciprocal cubes
0:32 The sum converges: watching the digits settle
0:55 Introducing ζ(3) — Apéry's constant
1:34 The Basel problem and Euler's π²/6
2:09 Beukers' double integral approach
3:05 The logarithmic fix — why the integral converges
3:38 Rotating coordinates to reveal π
4:05 Trying the same trick for cubes — hitting the wall
5:10 Even powers have closed forms; odd powers have nothing
5:59 Apéry's 1978 announcement and the skeptical reception
6:49 Cohen verifies the proof — every step holds
7:34 ζ(3) in QED — the electron's magnetic moment
8:49 What we still don't know — transcendence, closed forms, motivic cohomology
9:43 Closing — the simplest questions take the longest
Music:
All music by Vincent Rubinetti
Piece 1: Reflections
Piece 2: Trinkets
Piece 3: Resonance
#aperysconstant #riemannzetafunction #numbertheory #basel problem #euleridentity #quantumelectrodynamics #irrationalnumbers #motiviccohomology
