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Large Scale Structure of the Legendre Symbol
The following image is a much larger version of the table on the wikipedia page for the Legendre Symbol. I sort of like throwing big data at problems like this, or, to express it with greater clarity: the table on the wikipedia page is too small! source code tarball. Quadratic nonresidues are in red. Quadratic residues are in blue (1) or green (0). I wonder if the radial striping also appears in the Hilbert and Artin symbols? Probably could use the attention of a number theorist proper.

Bounty: I am offering a bottle of whisky to anyone who can successfully translate Jacob Lurie's Higher Topos Theory (and/or the stacks project) into the analytic combinatorical language of the manner of Flajolet's Analytic Combinatorics.