Relaxation to equilibrium and the spacing distribution of zeros of the Riemann ζ function

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Abstract

A novel approach is proposed to the distribution of spacings between zeros of the Riemann zeta function. Starting from the observation that the spacing distribution for zeros near the real axis is sharper than the asymptotic distribution, and that all computed moments grow monotonically as zeros are computed farther and farther away from (z) ≤ 0, an analogy with relaxation to equilibrium in a statistical system is drawn. Namely, it is conjectured that the spacing distribution evolves in a fictitious time , where T is the zeros' imaginary part, in the same way as the eigenvalues of a random matrix with mixed GUE and GSE symmetries. The time evolution restores asymptotically (at T → ∞) the GUE symmetry. Dyson's Brownian motion model for the eigenvalues of a random matrix is used for describing the time evolution, and an approximate, analytic description of the spacing distribution is conjectured, valid to first order in exp(-t).

Original languageEnglish (US)
Article number009
Pages (from-to)5893-5903
Number of pages11
JournalJournal of Physics A: Mathematical and Theoretical
Volume40
Issue number22
DOIs
StatePublished - Jun 1 2007

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

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