Logarithmic moments of characteristic polynomials of random matrices

In a recent article we have discussed the connections between averages of powers of Riemann’s ζ-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for ζ-functions, and the correspondence is again striking. Laboratoire de Physique Théorique de l’École Normale Supérieure, Unité Mixte de Recherche 8549 du Centre National de la Recherche Scientifique et de l’École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France. [email protected] † Department of Basic Sciences, University of Tokyo, Meguro-ku, Komaba 3-8-1, Tokyo 153, Japan. [email protected] 1 Correlation functions of characteristic polynomials We first briefly review the result of a previous paper [1], in which we have investigated the average of a product of characteristic polynomials of a random matrix. Let X be an M × M random Hermitian matrix. The correlation function of 2K distinct characteristic polynomials is defined as F2K(λ1, · · · , λ2K) = 〈 2K