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 distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N → ∞ asymptotics of the moments of |Z| and Z/Z * . These CUE results are then compared with what is known about the Riemann zeta function ζ(s) on its critical line Res = 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T . Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2 + iT ) in the limit T → ∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T . This leads us to a conjecture for the moments of |ζ(1/2 + it)|. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.

Abstract. 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 form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined 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.

random matrix theory, number theory, L-functionsRecent results of Katz and Sarnak [9,10] suggest that the lowlying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U (N ), O(N) or U Sp (2 N ). We here explore the link between the value distributions of the L-functions within these families at the central point s = 1/2 and those of the characteristic polynomials Z (U, 3 ) of matrices U with respect to averages over SO (2N) and U Sp ( 2N) at the corresponding point 3 = 0, using techniques previously developed for U (N) in [7]. For any matrix size N we find exact expressions for the moments of Z (U, 0 ) for each ensemble, and hence calculate the asymptotic (large N ) value distributions for Z (U, 0) and log Z (U, 0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of Lfunctions at the central point.

Abstract.Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP .

We argue that the freezing transition scenario , previously conjectured to occur in the statistical mechanics of 1/ f -noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p N ( θ ) of large N × N random unitary (circular unitary ensemble) matrices U N ; i.e. the extreme value statistics of p N ( θ ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μ N ( x ) of the intervals in which | p N ( θ )|> N x , x >0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ ( s ) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

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We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians -those that are related to quadratic forms of Fermi operators -between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N → ∞. This is shown to grow logarithmically with N . The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N×N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function ζ(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random-matrix theory, and the theory of the Riemann zeta function.