Random matrix theory and discrete moments of the Riemann zeta function

2002/07/31 by C. P. Hughes · 31 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Arithmetic zeta function #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Particular values of Riemann zeta function #Physics #Prime zeta function #Proof of the Euler product formula for the Riemann zeta function #Pure mathematics #Quantum mechanics #Random matrix #Riemann Xi function #Riemann hypothesis #Riemann zeta function #Unitary matrix #Unitary state #math.NT

paper · pdf · doi:10.1088/0305-4470/36/12/303

published in Journal of Physics A Mathematical and General 36(12), 2907-2917 (Institute of Physics) · Replacement corrects slight typos. To be published in J. Phys. A, special issue in Random Matrix Theory

arxiv created 2002/12/04 · openalex publication_date 2003/03/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding moments of the zeta function and its derivative.

Citations

• Mean-Value Theorems in the Theory of the Riemann Zeta-Function
• Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes
• Random Matrix Theory and ζ(1/2+ it)