• Corpus ID: 22258

Moments of the derivative of the Riemann zeta-function and of characteristic polynomials

@article{Conrey2005MomentsOT,
  title={Moments of the derivative of the Riemann zeta-function and of characteristic polynomials},
  author={J. Brian Conrey and Michael O. Rubinstein and Nina C. Snaith},
  journal={arXiv: Number Theory},
  year={2005}
}
We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann zeta-function on the critical line. We do the same for the analogue of Hardy's Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel… 
2 Citations
Random matrix theory and the sixth Painlevé equation
A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realized by matrices with Gaussian
Boundary conditions associated with the Painlevé III′ and V evaluations of some random matrix averages
In a previous work a random matrix average for the Laguerre unitary ensemble, generalizing the generating function for the probability that an interval (0, s) at the hard edge contains k eigenvalues,

References

SHOWING 1-10 OF 24 REFERENCES
A Hybrid Euler-Hadamard product formula for the Riemann zeta function
We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the
Random matrix theory and discrete moments of the Riemann zeta function
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
Integral Moments of L‐Functions
We give a new heuristic for all of the main terms in the integral moments of various families of primitive L‐functions. The results agree with previous conjectures for the leading order terms. Our
Random matrix theory and the zeros of ζ′(s)
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
Autocorrelation of Random Matrix Polynomials
Abstract: We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and
Characteristic polynomials of random matrices at edge singularities
  • Brézin, Hikami
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 2000
TLDR
There are remarkably simple formulas for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.
Zeros of Derivatives Of the Riemann Zeta-Function Near the Critical Line
The question of the horizontal distribution of the zeros of derivatives of Riemann’s zeta-function is an interesting one in view of its connection with the Riemann Hypothesis.
Random Matrix Theory and ζ(1/2+it)
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
On the Characteristic Polynomial¶ of a Random Unitary Matrix
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
On the zeros of ?'(s near the critical line
Let ρ = β ′ + i γ ′ denote the zeros of ζ (s), s = σ + i t . It is shown that there is a positive proportion of the zeros of ζ (s) in 0 < t < T satisfyingβ ′ − 1/2 (logT)−1. Further results relying
...
...