Random Matrix Theory and ζ(1/2+it)
@article{Keating2000RandomMT, title={Random Matrix Theory and $\zeta$(1/2+it)}, author={Jonathan P. Keating and Nina C. Snaith}, journal={Communications in Mathematical Physics}, year={2000}, volume={214}, pages={57-89} }
Abstract: We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit…
671 Citations
Random matrix theory and the zeros of ζ′(s)
- Mathematics
- 2003
We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on…
Random Matrix Theory and L-Functions at s= 1/2
- Mathematics
- 2000
Abstract: Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N),…
The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios
- Mathematics
- 2014
We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a…
On the characteristic polynomial of the eigenvalue moduli of random normal matrices
- Mathematics
- 2022
We study the characteristic polynomial p n ( x ) = Q n j =1 ( | z j | − x ) where the z j are drawn from the Mittag-Leffler ensemble, i.e. a two-dimensional determinantal point process which…
On the moments of the moments of $\zeta(1/2+it)$
- Mathematics
- 2020
Taking t at random, uniformly from [0, T ], we consider the kth moment, with respect to t, of the random variable corresponding to the 2βth moment of ζ(1/2 + ix) over the interval x ∈ (t, t + 1],…
CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES
- Mathematics
- 2009
We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent…
On the Characteristic Polynomial¶ of a Random Unitary Matrix
- Mathematics
- 2001
Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at…
Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles
- Mathematics
- 2014
We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral), and Jacobi β-ensembles of N×N random matrices. More specifically, we calculate scaling limits of the…
Symmetry Transitions in Random Matrix Theory & L-functions
- Mathematics
- 2008
We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N →…
References
SHOWING 1-10 OF 49 REFERENCES
Random Matrix Theory and L-Functions at s= 1/2
- Mathematics
- 2000
Abstract: Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N),…
Characteristic Polynomials of Random Matrices
- Mathematics
- 2000
Abstract: Number theorists have studied extensively the connections between the distribution of zeros of the Riemann ζ-function, and of some generalizations, with the statistics of the eigenvalues of…
On the Characteristic Polynomial¶ of a Random Unitary Matrix
- Mathematics
- 2001
Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at…
Random matrix theory and the Riemann zeros. I. Three- and four-point correlations
- Mathematics
- 1995
The non-trivial zeros of the Riemann zeta-function have been conjectured to be pairwise distributed like the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory.…
Mean-values of the Riemann zeta-function
- Mathematics
- 1995
Let Asymptotic formulae for I k ( T ) have been established for the cases k =1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of I k ( T ) remains…
The Riemann Zeros and Eigenvalue Asymptotics
- MathematicsSIAM Rev.
- 1999
It is speculated that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP, and very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues.
Semiclassical formula for the number variance of the Riemann zeros
- Mathematics
- 1988
By pretending that the imaginery parts Em of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it…
Gaussian fluctuation in random matrices.
- MathematicsPhysical review letters
- 1995
This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results on chaotic quantum systems and on the statistics of zeros of the Riemann zeta function.
Level spacings distribution for large random matrices: Gaussian fluctuations
- Mathematics
- 1998
We study the level-spacings distribution for eigenvalues of large N X N matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1.…
Periodic Orbits, Spectral Statistics, and the Riemann Zeros
- Mathematics
- 1999
My purpose in this article is to review the background to some recent developments in the semiclassical theory of spectral statistics. Specifically, I will concentrate on approaches based on the…