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Zeta function
For real s the Riemann zeta function Zeta[s] is given by Sum[1/n s , {n, ∞ }] or Product[1/(1 - Prime[n] s ), {n, ∞ }] . … 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 .
… In 1972 Sergei Voronin showed that Zeta[z + (3/4 + t)] has a certain universality in that there always in principle exists some t (presumably in practice usually astronomically large) for which it can reproduce to any specified precision over say the region Abs[z] < 1/4 any analytic function without zeros.

Many of these are related to the so-called Riemann zeta function, a version of which is shown in the picture below.
… A curve associated with the so-called Riemann zeta function. … The curve shown here is the so-called Riemann–Siegel Z function, which is essentially Zeta[1/2 + t] .