Off = Ptolemy, pegged at the origin 0, where the walk starts. On = pegged at
the limit of the series, ζ(s), which is what the late terms coil round. At one s
this only slides the picture; the shape is the same spiral. What changes is what
stands still when s moves: pegged at 0 the spiral's centre wanders along
ζ(s); pegged at ζ(s) the centre stays put and the starting point wanders instead.
At a zero this switch does nothing, correctly — ζ(s)=0
there, so the Sun already sits on the origin. The app boots on zero #1, so move t
off it (or use ?t=30) before judging the effect.
The Earth is the point the walk was pegged to before:
S0 = 0, the
empty sum, where the first step leaves from. Seen from the Sun it sits at
−ζ(s), at distance |ζ(s)|, and as s moves it traces −ζ(s) itself — the
values of ζ along your path through the s-plane, turned half a circle about the
Sun. It passes through the Sun exactly at a zero. With co-rotating θ on the
critical line it moves on a straight line: it sits at −Z(t), Hardy's function.
Drawn as a green rod dropped onto the plane, a ring where it lands, and its orbit
along the plane; pegged at 0 it stands still, as Ptolemy's Earth does.