Random Matrix Theory Predictions for the Asymptotics of the Moments of the Riemann Zeta Function and Numerical Tests of the Predictions

In 1972, H.L. Montgomery and F. Dyson uncovered a surprising connection between the Theory of the Riemann Zeta function and Random Matrix Theory. For the next few decades, the major developments in the area were the numerical calculations of Odlyzko and conjectures for the moments of the Riemann Zeta function (and other L-functions) found by Conrey, Ghosh, Gonek, Heath-Brown, Hejhal and Sarnak. Recently, there have been two important advances. First, Keating and Snaith, in a 2000 paper, conjectured connections between the moments of the characteristic polynomials of random matrices and the moments of the Riemann Zeta function. Second, Katz and Sarnak proposed connections between certain families of L-functions and other matrix groups. Our goal in this paper is twofold. First, we discuss links between Random Matrix Theory and the Zeta function. Then, we describe our numerical calculation of the moments of the Zeta function and compare initial results with Random Matrix Theory predictions.