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Orthogonal and symplectic $n$-level densities

About this Title

A. M. Mason and N. C. Snaith

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 251, Number 1194
ISBNs: 978-1-4704-2685-9 (print); 978-1-4704-4262-0 (online)
Published electronically: September 11, 2017

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Eigenvalue Statistics of Orthogonal Matrices
  • Chapter 3. Eigenvalue Statistics of Symplectic Matrices
  • Chapter 4. $L$-functions
  • Chapter 5. Zero Statistics of Elliptic Curve $L$-functions
  • Chapter 6. Zero Statistics of Quadratic Dirichlet $L$-functions
  • Chapter 7. $n$-level Densities with Restricted Support
  • Chapter 8. Example Calculations


In this paper we apply to the zeros of families of -functions with orthogonal or symplectic symmetry the method that Conrey and Snaith (Correlations of eigenvalues and Riemann zeros, 2008) used to calculate the -correlation of the zeros of the Riemann zeta function. This method uses the Ratios Conjectures (Conrey, Farmer, and Zimbauer, 2008) for averages of ratios of zeta or -functions. Katz and Sarnak (Zeroes of zeta functions and symmetry, 1999) conjecture that the zero statistics of families of -functions have an underlying symmetry relating to one of the classical compact groups , and . Here we complete the work already done with (Conrey and Snaith, Correlations of eigenvalues and Riemann zeros, 2008) to show how new methods for calculating the -level densities of eigenangles of random orthogonal or symplectic matrices can be used to create explicit conjectures for the -level densities of zeros of -functions with orthogonal or symplectic symmetry, including all the lower order terms. We show how the method used here results in formulae that are easily modified when the test function used has a restricted range of support, and this will facilitate comparison with rigorous number theoretic -level density results.

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