Two many numbers..................

Numerical calculation by Odlyzko[edit]

The real line describes the two-point correlation function of the random matrix of type GUE. Blue dots describe the normalized spacings of the non trivial zeros of Riemann zeta function, the first 10⁵ zeros.

In the 1980s, motivated by the Montgomery's conjecture, Odlyzko began an intensive numerical study of the statistics of the zeros of ζ(s). He confirmed that the distribution of the spacings between non-trivial zeros using detail numerical calculation and demonstrated that the Montgomery's conjecture would be true and the distribution would agree with the distribution of spacings of GUE random matrix eigenvalues using Cray X-MP. In 1987 he reported the calculations in the paper Andrew Odlyzko (1987).

For non trivial zero, 1/2+iγ_n, let the normalized spacings be

Then we would expect the following formula as the limit, M, N →∞. Then

Based on a new algorithm developed by Odlyzko and Schönhage that allowed them to compute a value of ζ(1/2 + it) in an average time of t^ε steps, Odlyzko computed millions of zeros at heights around 10²⁰ and gave some evidence for the GUE conjecture.^[1][2]

The figure contains the first 10⁵ non-trivial zeros of the Riemann zeta function. As more zeros are sampled the more closely their distribution approximates the shape of the GUE random matrix.