Abstract:

Mahler's measure of a monic polynomial $f(x)$ is defined as the product of the absolute values of the roots of $f(x)$ that lie outside the unit disk. In 1933, D.~H. Lehmer asked if it was possible to find polynomials with integer coefficients having Mahler's measure arbitrarily close to~1, but greater than~1. This problem remains open, and it has several interesting consequences. In this talk, we will summarize some known facts and recent work regarding Mahler's measure and related topics, discuss some applications in number theory and ergodic theory, and present several open problems. If time permits, we will also mention some formulas recently conjectured by Boyd expressing certain limiting values of Mahler's measure in terms of values of $L$-functions associated with elliptic curves.