2013/07/05 by E. Bogomolny, J. P. Keating, Jonathan P. Keating
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Analytic Number Theory Research #Eigenvalues and eigenvectors #Explicit formulae #Hermitian matrix #Limit point #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Riemann hypothesis #Riemann zeta function #Universality (dynamical systems) #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/46/30/305203
published as J. Phys. A: Math. Theor. 46, 305203 (2013) · 22 pages
openalex publication_date 2013/07/05 · arxiv created 2013/07/23 · arxiv updated 2015/06/16 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
We illustrate a general method for calculating spectral statistics that combines the universal (random matrix theory limit) and the non-universal (trace-formula-related) contributions by giving a heuristic derivation of the three-point correlation function for the zeros of the Riemann zeta function. The main idea is to construct a generalized Hermitian random matrix ensemble whose mean eigenvalue density coincides with a large but finite portion of the actual density of the spectrum or the Riemann zeros. Averaging the random matrix result over remaining oscillatory terms related, in the case of the zeta function, to small primes leads to a formula for the three-point correlation function that is in agreement with results from other heuristic methods. This provides support for these different methods. The advantage of the approach we set out here is that it incorporates the determinantal structure of the random matrix limit.