The Riemann Hypothesis
Every non-trivial zero of ζ lies on the critical line
The Riemann zeta function ζ(s) = 1 + 1/2ˢ + 1/3ˢ + 1/4ˢ + … extends to the whole complex plane. The zeros that aren't at the negative even integers all seem to sit on a single vertical line: real part exactly 1/2. Bernhard Riemann conjectured this in 1859 and nobody has proved or disproved it since. A proof would lock down how prime numbers are distributed; one of the seven Millennium Prize Problems, $1M reward.
t_max — how far up the critical line
45.0t ∈ [0, 45.0]
The vertical reach of the ζ trace. Each loop through the origin marks one non-trivial zero. ~14 reveals the first; 60 reveals about 15.
N — Dirichlet series terms
200terms
ζ is computed as Σ_{n=1..N} (−1)^{n+1}/n^s scaled by 1/(1−2^{1−s}). 200 terms is more than enough for t ≤ 60.
N_zeros — zeros used in π reconstruction
8zeros
Number of non-trivial zeros plugged into Riemann's explicit-formula correction. 0 = the bare logarithmic integral Li(x). More zeros = the smooth curve climbs onto the staircase.
What you are seeing
Top plot: the staircase π(x) jumps by 1 at every prime, and the warm curve is the Riemann approximation reconstructed from the first N_zeros non-trivial zeros. With zero zeros, the curve is Li(x), the leading-order smooth estimate. As N_zeros grows the corrections ring the curve into the staircase — that is, literally, how the zeros encode the primes.
Bottom plot: the image of ζ along the critical line s = ½ + it, traced in the complex plane. Every loop through the origin is one zero. The hypothesis says this picture stays exactly the same picture forever — there is no t at which the orbit drifts off and avoids the origin while the real part of s sits anywhere else than ½.
