We study the characteristic polynomials Z(U,#)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 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

Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollified mean–square of the L–functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes–Vignéras Γ2–function and to a family of self–similar functions. 1.

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.

In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zeta-function on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We develop a new heuristic method to conjecture the asymptotic size of both the sixth and eighth moments. Our estimate for the sixth moment agrees with and strongly supports, in a sense made clear in the paper, the one conjectured by Conrey and Ghosh. Moreover, both our sixth and eighth moment estimates agree with those conjectured recently by J. Keating and N. Snaith based on modeling the zeta-function by characteristic polynomials of random matrices from the Gaussian unitary ensemble. Our method uses a conjectural form of the approximate functional equation for the zeta-function, a conjecture on the behavior of additive divisor sums, and D. Goldston and S. Gonek’s mean value theorem for

this paper, we present evidence to support the

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements, and a number of applications.

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. 1.

Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivated by applications to analytic number theory. Our approach uses classical methods of random matrix theory, in particular determinants and orthogonal polynomials, and can be considered more elementary than the method of Howe pairs used in the previous study. 1

We give a newheuristic 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 form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are de ned by the appropriate group averages. This lends support to the idea that arithmetical L-functions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

The purpose of the present article is to survey some mean value results obtained recently in zeta-function theory. We do not mention other important aspects of the theory of zeta-functions, such as the distribution of zeros, value-distribution, and applications to number theory. Some of them are probably treated in the articles of Professor Apostol and Professor Ramachandra in the present volume. Even in the mean value theory, we do not discuss many important recent topics. Those include: Recent progress in the theory of large values and fractional moments made by Heath-Brown [55] and the Indian school (Ramachandra, Balasubramanian, Sankaranarayanan and others, see Ramachandra [172]); mean values taken at the zeros or at the points near the zeros (Gonek [30] [31], Fujii [23]-[26] and others); the mean square of the product of the zeta-function and a Dirichlet polynomial (see Conrey-Ghosh-Gonek [19] and the papers quoted there). All of these three topics are closely connected with the distribution of zeros of zeta-functions, hence the full account of them would require too many pages. We will only discuss the theory of Titchmarsh series very briefly in Section 7. In the fourth power moment theory there have been remarkable developments which may be characterized by the use of the spectral theory of Maass wave forms. We mention this