The s-Plane

Every picture of the Riemann zeta function starts on this plane. Pick an input s on the left and see where zeta sends it on the right.

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ⋯ with s = σ + it. σ is how far right, t is how far up.

Map
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Input the s-plane

critical line σ = ½ zero ×pole at 1 vertical line through s horizontal line through s
Drag to move s. Scroll to zoom. Shift-drag or two fingers to pan.

Output where ζ(s) lands

The teal curve is where the whole vertical line through s lands. The rose curve is the horizontal line. When ζ(s) sits on the gold ring at 0, s is a zero.
Input s
Output ζ(s)
Size and angle
Where s is
Nearest zero

Size of ζ up the vertical line

the line through s the ½ line, for comparison
Each dip is the curve above passing near 0. On the ½ line the dips reach the floor. Drag here to change t.

Where are the primes? built from the zeros, one wave each

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Ten things to try

    Start anywhere

    Each step moves s to a spot worth seeing and switches on the views that show it best. The last one runs the primes panel. After that the controls are yours.

    What is proven, and what is not

    • ProvenEvery zero other than −2, −4, −6, … lies inside the strip 0 < σ < 1. Hadamard and de la Vallée Poussin, 1896. This is the fact that gives the prime number theorem.
    • ProvenThe zeros are mirrored across the ½ line and across the real axis. Riemann, 1859.
    • ProvenInfinitely many zeros sit exactly on the ½ line. Hardy, 1914. More than two fifths of all zeros do: Conrey, 1989.
    • CheckedEvery zero up to height t = 3 × 10¹² is on the ½ line. That is about 12 trillion zeros. Platt and Trudgian, 2021.
    • OpenAll of them are on the ½ line. This is the Riemann Hypothesis.

    Names for the parts

    s-plane
    The plane of inputs. Each point is one complex number s = σ + it. Also called the complex plane.
    Critical strip
    The band between σ = 0 and σ = 1. All the zeros that matter for primes live here.
    Critical line
    The line σ = ½ up the middle of the strip.
    Pole
    The point s = 1, where ζ has no value because the sum grows forever.
    Trivial zeros
    The zeros at −2, −4, −6, … on the real axis.
    Why primes
    Right of σ = 1, ζ(s) equals a product with one factor for each prime. Each zero in the strip adds one wave to the count of primes. Its height t is the wave's pitch and its σ sets how large the wave grows. Zeros at σ = ½ give the smallest waves possible. The panel “Where are the primes?” adds those waves up.

    Values are computed in your browser to about ten digits, using Euler–Maclaurin summation right of σ = 0 and the mirror formula left of it. Zeros are located on the ½ line up to height 1000. The primes panel is the explicit formula with tapered (Fejér) weights, so each bump narrows smoothly as zeros are added.