Mahler's Measure and Special Values of L-functions

If P(Xl,… xn) is a polynomial with integer coefficients, the Mahler measureM(P) of P is defined to be the geometric mean of |P| over the n-torus Tn. For n = 1, M(P) is an algebraic integer, but for n > 1, there is reason to believe that M(P) is usually transcendental. For example, Smyth showed that log M(l + x +y) = L'(X–3, −1), where X–3 is the odd Dirichlet character of conductor 3. Here we will describe some examples for which it appearsthat log M(P(x, y)) = rL'(E, 0), where E is an elliptic curve and r is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.