We prove a multidimensional extension of Selberg's central limit theorem for log ζ, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence n ↔ log t not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.
- Central limit theorem
- Zeta and L-functions
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty