Pictures that belong to no paper

Four panels. The first zero watched as the continuation is undone — it never leaves the line. The zero set at four stages, the irregular zeros of ζ against a uniform comb. The ruler law, with the comb's pitch at 2π/log X. And the walk itself, drawn, which is what the question behind the picture was actually about.

The claim tested here was the author's, made from a screenshot of the walk: that the spikes along the arm are artefacts of overlaid mirrored rectangles — an eight-spike star. The picture asks two things of it, and the second is the one that could have refuted him: are the spikes real and periodic, and does a control object show them too. Conductor 5.

Every step rescaled to unit length with its direction untouched, for four objects at one height in one window: ζ, a complex character of order 4, a real character as the control, and the counterexample. The decay in the step lengths is what makes the walk a spiral, so removing it leaves the skeleton.

The walk is split at the term where the turn per step passes π. Below it lies all of the curl; above it a smooth spiral. The finding printed on the picture is that the chirality flip is an artefact of the rescaling and the curl is not. Two further panels: measured in log n the rotation rate is exactly constant, so the apparent slowdown is a reparametrisation and nothing else; and σ never enters the angles at all — it sets only the step lengths, so the term where the split happens is the same for every σ.

This is the picture that prompted this page.

The same objects, asking what a local reading of the walk actually sees. The turn per step falls monotonically and never reverses. What a local reading sees instead is the sign flipping — 220 apparent reversals in all, every one of them below term 318, and beyond that nothing flips again. The sign flips; the motion never does.

A pair of zeros that merges and leaves the critical line as a parameter is varied, tracked across heights and truncations with the counterexample as the control. The quantity plotted is how much of the remainder is still unspent at the moment an off-line pair lands on the line.

This is the deformation picture, and it is a different object from the static near-collision statistic of the ninth paper — that paper's negative result is explicit that it does not touch this.

The previous control held the gamma factor's parity fixed and let the conductor float, comparing the conductor-5 counterexample against Euler products at conductors 3 and 7. That brackets the conductor without matching it — and the conductor itself turned out to carry a signal among the Euler products, so it was live noise inside the very comparison it was meant to clean. This is the same test with the conductor matched exactly and the objects paired.

Six panels on the vector from the origin to the partial sum. The tether normalised by arc length, in a coherent deep window, in the finite half, and deeper again. The depth at which the spiral begins to exist at all — about twenty times the Riemann–Siegel cut, before which it simply is not there. A degree-0 object drawn three ways from an identical multiset of step lengths, where only the arrangement differs. And the waviness of each tether, window by window.

Built from one sentence of the author's: that the tethers of the terms taking part in locating the limit axis are perfectly smooth in the complex plane, without needing to be normalised at all. The picture builds exactly that function — the distance from the origin to the partial sum, as a function of the term index, raw, with nothing divided by anything — and builds nothing else. No object comparison, no coefficient statistic, no summary number.

The same length for every object this line has a star shape for. The length settles, and what it settles on is the modulus of the L-function at that height: 50 readings across several heights, worst absolute error 4.8e−09.

ζ is the one object whose length runs away instead of settling, and the picture names the reason: it is the only one of them whose coefficients do not sum to zero over a period. That surplus is its pole.

Measured from the axis rather than from the origin, and normalised by the logarithmic shrinking. It is a correction: the figures before it had measured from the wrong point, and the near-constant length the author expected only appears once the measurement is taken where he was taking it.

Past the last chirality switch the remainder is a regular spiral, and the distance of the term vertices from the origin is the slow, smooth, rising curve. The panels also separate the two cases that matter: measured against the bare spiral radius the departure shrinks at a zero and does not shrink off one.

The same measurement in the same order for the three objects beside ζ — a complex character of order 4, a real character, and the counterexample, the last of which has two distinct segment lengths rather than one. The falling law holds off the critical line as well as on it, with exponent 1−σ in place of ½.

A correction to the picture above it. That one plotted every two-hundredth term, and the structure being asked about has period 5 — it had been aliased away, and the author caught it. Here every term is drawn.

The picture also carries a correction against its own earlier reading, printed on it rather than quietly fixed: one of the objects has three distinct strand heights, not four. The two classes that read as different are equal, and they converge more slowly than the others — a convergence drift had been read as a distinction.

Past the leading law, is there anything left moving, and if so does it oscillate or does it drift smoothly? Asked of all four objects, at their own zeros, at every term.

The question the picture above answered was not the question that had been asked. The author's own words were that at ζ the consecutive terms lie on a straight line and at every other object they go up and down — and the previous answer had removed the period-q law first and then reported that no oscillation was left, which is a different statement entirely. The law removed was the oscillation.

So this picture draws it. The oscillation is the running partial sums of the coefficient list, and nothing more.

Its ceiling, printed with it. This is a property of the coefficient list, so it cannot on its own separate ζ from the rest at degree 1 without reducing to the hypothesis itself. What it does separate is ζ from every object whose coefficients sum to zero over a period — and that is the pole again, not the Euler product. The counterexample sits with the characters here, on the same side as the objects that keep every zero on the line.

The author's conjecture, in his own words: that ζ is the natural expression of the smoothness of the primes and everything else is a deviation from it, as though ζ were π drawing a perfect circle and the other objects were drawing ellipses.

The picture runs the test that conjecture actually asks for, which is not ζ against unrelated L-functions — it is ζ against objects that share its zeros exactly and differ elsewhere. The result is on the face of it: with the pole kept, the de-scaled walk at ζ's own zero is a circle. With the pole destroyed, it is a ring with lobes.

Its ceiling, printed with it. Having a pole is a property of the coefficient list, and at degree 1 with a functional equation the only objects that have one are ζ and its shifts — so "smooth if and only if ζ" reduces to the hypothesis before it is measured. What the picture is worth is as a description: it says exactly which deformation destroys the smoothness, and it says that moving the zeros does not.