相关论文: Logarithmic moments of characteristic polynomials …
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…
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…
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…
Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those…
We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the…
Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $\zeta(s)$. Their approach indicates how lower twisted moments of $\zeta(s)$ may be used…
We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…
For random matrix ensembles with unitary symmetry, there is interest in the large $N$ form of the moments of the absolute value of the characteristic polynomial for their relevance to the Riemann zeta function on the critical line, and to…
We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor…
We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process…
Keating and Snaith showed that the $2k^{th}$ absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree $k^2$. In this article, uniform asymptotics for the…
We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both…
In this article, we give, under the Riemann hypothesis, an upper bound for the exponential moments of the imaginary part of the logarithm of the Riemann zeta function on the critical line. Our result, which gives information on the…
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 conjectures also have an almost identical…
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the…
We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried…
Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are…
In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity.…
Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…
In this article, we provide a unified framework for studying the convergence of rescaled characteristic polynomials of random matrices from various classical ensembles as well as functional convergence results for the Riemann zeta function.…