ζ Riemann Zeta Function — Critical Strip & Zeros

The plain series 1/1ⁿ + 1/2ⁿ + 1/3ⁿ + … only converges for Re(s) > 1, so it cannot be summed directly inside the strip. The code instead sums the alternating Dirichlet eta series η(s) = Σ(−1)ⁿ⁻¹ n⁻ⁿ for n from 1 up to the Terms value, which converges for Re(s) > 0, then divides by the factor (1 − 2¹⁻ⁿ) using complex division to recover ζ(s) = η(s) / (1 − 2¹⁻ⁿ).

Each pixel corresponds to a complex point s, and its colour is computed directly from ζ(s). The hue is set by the argument (angle) of ζ(s) in the complex plane, mapped across the full 360° colour wheel, while the lightness is a compressed function of the modulus |ζ(s)|: as |ζ(s)| shrinks toward zero the pixel darkens toward black, which is why zeros of ζ appear as small dark points where every hue meets.

The critical strip is the vertical band 0 < Re(s) < 1, shown in the simulation between two dashed lines. The critical line is the line Re(s)=½ running down its centre, drawn in solid white. The unproven Riemann hypothesis states that every non-trivial zero of ζ(s) lies exactly on this critical line rather than merely somewhere inside the strip.

In the "Critical line" view the point s=½+it climbs straight up the critical line as t increases, and the plot tracks the modulus |ζ(½+it)|. The simulation marks ten known non-trivial zeros at t ≈ 14.13, 21.02, 25.01, 30.42, 32.94, 37.59, 40.92, 43.33, 48.01 and 49.77; every time the walking point reaches one of these heights, |ζ| genuinely falls to zero because s is a root of the zeta function there.

The Dirichlet eta series is an infinite sum, and the simulation can only add up a finite number of terms (10 to 200, chosen by the slider) before stopping. With few terms the partial sum is a rougher approximation to the true ζ(s), which can slightly blur colour boundaries and zero positions; with more terms the sum converges closer to the exact value at the cost of more arithmetic per pixel, which is why higher Terms settings render more slowly.

Through its Euler product, ζ(s) equals an infinite product over all primes p of 1/(1−p⁻ⁿ), which is why its zeros are not arbitrary: Riemann's explicit formula expresses the exact count of primes up to any number as a smooth main term plus a correction built from oscillating terms, one per non-trivial zero. The precise locations of the zeros shown in this simulation therefore directly control how evenly or unevenly the primes are distributed.