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Table[{n, s, (Prime[n]^s/(Prime[n]^s - 1))}, {n, 1, 2000}, {s, 2, 80}], PlotRange -> All]  

enter image description here

Is the pattern the result of the data, the plotting function, or a bit of both?

Edit fixed bad exponents, replaced the plot.

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It seems to be Moiré pattern – Kuba Apr 17 '14 at 7:46
@Kuba, perfect! Make it an answer and I'll sign off. – Fred Kline Apr 17 '14 at 7:55
up vote 4 down vote accepted

What you see is Moiré pattern

Closely related topic with 2D case: Using high RasterSize changes contour pattern

Worth to add that the patterns does not seem to have a translation symmetry because the projection is not parallel. You can compare it with distant ViewPoint case:

 Table[{n, s, (Prime[n]^s/(Prime[n]^s - 1))}, {n, 1, 2000}, {s, 2, 
   80}], PlotRange -> All, ViewPoint -> 1000 {1, -1, 1}, 
 ImageSize -> 400]

enter image description here

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