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The picture in the main text shows RiemannSiegelZ[t] , defined as Zeta[1/2 +  t] Exp[  RiemannSiegelTheta[t]] , where RiemannSiegelTheta[t_] = Arg[Gamma[1/4 +  t/2]] - t Log[ π ]/2 The first term in an approximation to RiemannSiegelZ[t] is 2 Cos[RiemannSiegelTheta[t]] ; to get results to a given precision requires summing a number of terms that increases like √ t , making routine computation possible up to t ~ 10 10 .
To make a random walk on a lattice with k directions in two dimensions, one can set up e = Table[{Cos[2 π s/k], Sin[2 π s/k]}, {s, 0, k - 1}] then use FoldList[Plus, {0, 0}, Table[e 〚 Random[Integer, {1, k}] 〛 , {t}]] It turns out that on any regular lattice, in any number of dimensions, the average behavior of a random walk is always isotropic.
As discovered by Srinivasa Ramanujan in 1918 its fluctuations (see below) can be obtained from the formula 1/6 π 2 n Sum[Apply[Plus, Cos[2 π n Select[ Range[s], GCD[s, #]  1 &]/s]]/s 2 , {s, ∞ }] (c) Squares are taken to be of positive or negative integers, or zero.
Discretizing yields lattice gauge theories with energy functions involving for example Cos[ θ i - θ j ] for color directions at adjacent sites.
For large j i they are approximated by Cos[ θ + π /4]/Sqrt[12 π v] , where v is the volume of the tetrahedron and θ is a deficit angle.