Zeta Ball ζ

The zeta function painted on a ball of diameter ½ that rests on the plane at s = ½, with a lamp at its top. Its poles sit over 0 and 1, and its upright great circle casts the critical line. Drag the ball to turn it. The shadow panel shows what the lamp casts on the s-plane, computed live.

the balldrag to turn · lamp at the top
refining
its shadow on the s-planedrag to pan · scroll to zoom
refining
Painted with
Quarter turns about the ball's own axes
½ ↔ ∞ upright axis 0 ↔ 1 through the poles ½ ± i/2 front to back
Shadow view
±3
rotation q1
under the lamps = ∞
touching the floors = ½
From the plane phase to ζ

3b1b's grid for the analytically continued ζ: the half-plane Re s > 1 in warm colours, its mirror image in cool ones. Slide from the sphere in its plane phase (foci at −1 and 1, as in the quaternion explorable) through x ↦ (x + 1)/2 to this ball's own grid (foci 0 and 1), then on to ζ itself. The middle stop is 1/(1 − 2¹⁻ˢ), and ζ(s) = η(s)/(1 − 2¹⁻ˢ), so ζ is the ball's grid bent by η.

computing ζ
Things to try
  1. Paint with ξ, widen the shadow view to about ±40 so the zeros show, and give the ½ ↔ ∞ axis a quarter turn. The critical line and the real axis trade places. They are the only two lines through ½ where ξ is real: on one it swings like cos and holds the zeros, on the other it grows like cosh.
  2. Turn it once more. ξ looks exactly as it started, because ξ(s) = ξ(1 − s). With ζ the colours change.
  3. Turn about 0 ↔ 1. The point at infinity, where every zero piles up, comes down to ½ − i/2 and then onto the touching point. Press “Zoom to where ∞ landed” to see the zeros lined up into it.
  4. Turn about ½ ± i/2. The critical line closes into a circle around the pole, which now sits at the touching point.
  5. Four quarter turns about any axis bring the ball home, but q reads −1. Four more make it +1.
Key
critical line, Re s = ½ (i-circle)
real axis (k-circle)
waist |s − ½| = ½, through 0 and 1 (j-circle)
nontrivial zeros, the first 79 above and below
trivial zeros −2, −4, −6, … (zeta only; ξ has none)
∞, the point under the lamp at the start

Colour is the angle of the value, brightness its size (dark small, bright large). Grey marks points too far out to compute here. The ball's own coordinate z = (s − 1)/s puts its six marked points at z = 0, ∞, ±1, ±i.