The Mahler measure of a complex-valued polynomial P(z) is the geometric mean of the modulus of the polynomial evaluated over the unit circle. At first glance, this quantity appears to be a harmless integral, but it turns out that it invades several areas of number theory. In this talk, we will discuss the relationship between Mahler measure, the golden ratio and algebraic integers. We will also extend the definition of Mahler measure to multivariable functions, and will conclude with evaluating the Mahler measure of a polynomial in three variables, finding that it can be expressed in terms of the Riemann zeta function evaluated at 3.
Mahler measure