Zeta function
For real s the Riemann zeta function Zeta[s] is given by Sum[1/ns, {n, ∞}] or Product[1/(1 - Prime[n]s), {n, ∞}]. The zeta function as analytically continued for complex s was studied by Bernhard Riemann in 1859, who showed that PrimePi[n] could be approximated (see page 909) up to order √n by LogIntegral[n] - Sum[LogIntegral[n^r[i]], {i, -∞, ∞}], where the r[i] are the complex zeros of Zeta[s]. The Riemann Hypothesis then states that all r[i] satisfy Re[r[i]] 1/2, which implies a certain randomness in the distribution of prime numbers, and a bound of order √n Log[n] on PrimePi[n] - LogIntegral[n]. The Riemann Hypothesis is also equivalent to the statement that a bound of order √n Log[n]2 exists on Abs[Log[Apply[LCM, Range[n]]] - 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 ~ 1010.
It is known that:
• The average spacing between zeros decreases like 1/Log[t].
• The amplitude of wiggles grows with t, but more slowly than t0.16.
• At least the first 10 billion zeros have Re[s] 1/2.
The statistical distribution of zeros was studied by Andrew Odlyzko and others starting in the late 1970s (following ideas of David Hilbert and George Pólya in the early 1900s), and it was found that to a good approximation, the spacings between zeros are distributed like the spacings between eigenvalues of random unitary matrices (see page 977).
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.