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The first plot shows a large number of terms of Zeta(0.5 + i t) plotted end to end for t = 778948.517. The other plots are two zoomed-in regions, including one ending in a Cornu spiral. Despite all sorts of vicissitudes, the plot generally spirals outwards in a "purposeful" sort of way. It is not hard to find values of t that give such plots.

What causes this phenomenon?

Now for the first few terms, it can be explained as an aliasing effect where successive terms have approximately the same phase (angle) modulo 2 Pi. Similarly, the Cornu spiral is also an aliasing / sampling effect where the angle modulo 2Pi changes gradually.

But what about the emergence of a broad trend from a really long series of terms that seem to be changing randomly in angle?

What causes this phenomenon?

Now for the first few terms, it can be explained as an aliasing effect where successive terms have approximately the same phase (angle) modulo 2 Pi. Similarly, the Cornu spiral is also an aliasing / sampling effect where the angle modulo 2Pi changes gradually.

But what about the emergence of a broad trend from a really long series of terms that seem to be changing randomly in angle?

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