Count the primes below a trillion and you get 37,607,912,018. That number isn't random noise. It's a smooth curve plus an infinite stack of waves: one wave for each zero of a function Bernhard Riemann wrote down in 1859. The Riemann Hypothesis is just the conjecture that every one of those waves is in tune.
TIMESTAMPS:
0:00 The prime counting staircase — erratic up close, smooth from a distance
1:00 Gauss conjectures x/ln(x) as a teenager; the logarithmic integral Li(x)
2:05 Zooming in on the gap — it doesn't drift, it oscillates
2:44 Building the zeta function from scratch
3:43 Euler's product formula — the sum over integers equals a product over primes
4:29 Riemann lets s be complex; analytic continuation explained
5:29 ζ(−1) = −1/12 and what it actually means
6:05 The nontrivial zeros and the critical strip
7:25 The Riemann Hypothesis stated in one sentence
8:23 Riemann's exact formula for the primes — each zero is a wave
9:34 The primes are a chord; why the location of zeros controls their regularity
11:06 Connections to quantum mechanics, random matrix theory, and the unmapped web
Music:
All music by Vincent Rubinetti
Piece 1: Reflections
Piece 2: Trinkets
Piece 3: Resonance
Piece 4: Heartbeat
All animations produced using Manim. All credits there to 3B1B.
#riemannhypothesis #primenumbers #zetafunction #analyticcontinuation #numbertheory #complexanalysis #millenniumprize #eulerproduct
