Michael Rubinstein

L-functions

Michael Rubinstein works in analytic number theory, focusing his efforts on L-functions. L-functions are functions which encode arithmetic properties of a mathematical construct. They govern, for example, the distribution of prime numbers, the structure of elliptic curves and they provide information about many other problems in number theory. “L-functions are poorly understood,” explains Michael. “Basic properties are not established, answers to fundamental questions about their behaviour are not known. So I’ve been using models to make predictions for L-functions. It involves a lot of probability – we use the models to make guesses about the correct behaviour and then test them numerically on the computer.”

One of the behaviours Michael explores deals with the zeros of the Riemann zeta function; at which complex values does this function equal zero. To explore this, Michael is using random matrix theory. “There are ways to interpret matrices as functions. What we’re comparing,” explains Michael. “Is not the L-function to the matrix, but the L-function to the characteristic polynomial of the matrix. That random matrices should have anything to do with number theory remains somewhat of a mystery and suggests a deep theory waiting to be uncovered.”

Random matrix theory gives Michael some intuitions about L-functions, which he then explores numerically. “Globally they share properties, but if you look more closely there’s a difference. We managed – just from the inspiration provided by random matrix theory – to make guesses as to what happens on the number theory side by adding an extra layer of arithmetic,” Michael notes. “Statistically, L-functions exhibit the same properties as the characteristic polynomials of certain kinds of large matrices, specifically unitary matrices.”

University of Waterloo Mathematics, Annual Report 2006