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maths · two poles · zeta ball · strut · the dial
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The zeta dial

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Turn the ball about the strut through ½ and ∞. On the plane that is a plain rotation about ½: s − ½ ↦ eiθ(s − ½). A quarter turn carries the real line onto the critical line (the Wick rotation); a half turn is the functional equation s ↦ 1 − s; a full turn is the identity, with q = −1. The paint shows the function pulled back through the turn, so you watch cosh become cos.

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ξ pulled back through the turn

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white line: where the real axis has gone · gold: where the critical line has gone · dots: the zeros, carried along

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painting
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The ball, from above the lamp

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looking down the ½–∞ strut: the equator is the rim, the real axis and the imaginary axis are the two diameters; the needle is θ

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+ θ0 + on the planes − ½ ↦ s − ½ + q = e^{iθ/2}1 + even sheet sees+1 + odd sheet sees+1 + F seesu ↦ u +
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Three sheets, one dial

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F(u). Write v = s − ½. ξ(½ + v) = F(v²), and F has positive Taylor coefficients. F lives in the u = v² plane; the dial turns u at double speed, u ↦ e2iθu. A half turn of the ball is a full turn of F. The zeros of ξ, at v = ±iγ, become the single points u = −γ² on F's negative real axis. RH, downstairs: every zero of F is real and negative.

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ξ, even. Take the square root: v = √u. Now the half turn exists as its own move, s ↦ 1 − s, and ξ doesn't notice it: ξ(1 − s) = ξ(s). The two roots ±iγ are the zero pair on the equator.

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(s − ½)·ξ, odd. Multiply by v. Same zeros, plus a new one at v = 0, the touching point ½. The half turn now shows up as a sign: this sheet sees s ↦ 1 − s as −1. A full turn is +1 again on both sheets; the −1 that a full turn carries in the quaternion reading, q = eiθ/2, is what the odd sheet shows at the half.

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So the dial reads: 0 = nothing; ¼ = Wick, real axis onto the critical line, cosh onto cos; ½ = the functional equation (+1 even, −1 odd); 1 = home with q = −1; 2 = home with q = +1.

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Reading the paint

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Hue is the argument of the value, brightness its size. At θ = 0 the real axis is a bright cosh ridge and the critical line a cos stripe with dark zeros. Drag θ: the ridge and the stripe rotate together about ½. At ¼ they have swapped places on the screen. At ½ the picture is back (even sheet) or back with every hue rotated by half a turn (odd sheet, a sign flip).

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Zeros are the first 20 pairs on the critical line; on the F sheet they appear once each, on the negative real u-axis.

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