THE MAHLER MEASURE OF A THREE-VARIABLE FAMILY AND AN APPLICATION TO THE BOYD–LAWTON FORMULA JARRY GU AND MATILDE LAL ´ IN Abstract. We prove a formula relating the Mahler measure of an infinite family of three- variable polynomials to a combination of the Riemann zeta function at s = 3 and special values of the Bloch–Wigner dilogarithm by evaluating a regulator. The evaluation requires two different applications of Jensen’s formula and analyzing the integral in two different planes (as opposed to only one plane as usually). The degrees of the monomials involving one of the variables is allowed to vary freely, leading to an interesting application of the Boyd–Lawton formula. 1. Introduction The (logarithmic) Mahler measure of a non-zero rational function P ∈ C(x 1 ,...,x n ) is defined by m(P ) := 1 (2πi) n  T n log |P (x 1 ,...,x n )| dx 1 x 1 ··· dx n x n , where the integration is taken over the unit torus T n = {(x 1 ,...,x n ) ∈ C n : |x 1 | = ··· = |x n | =1} with respect to the Haar measure. This construction originated for one-variable polynomials in the search for large prime numbers (for example, see Lehmer’s work [Leh33]) and was later extended to multi-variable polynomials by Mahler [Mah62] in applications to classical heights of polynomials. Even- tually Mahler measure was found to yield special values of functions of number theoretic significance, such as the Riemann zeta function and other L-functions. The first examples of these relationships were given by Smyth [Smy81, Boy81b] m(x + y + 1) = 3 √ 3 4π L(χ −3 , 2), m(x + y + z + 1) = 7 2π 2 ζ (3), where L(χ −3 ,s) is the Dirichlet L-function in the character of conductor 3 and ζ (s) is the Riemann zeta function. Deninger explained the appearance of L-functions in some Mahler measure formulas in terms of Beilinson’s conjectures via relationships with regulators in [Den97]. (Additional insight into this direction can be found in the works of Boyd [Boy98] and Rodriguez-Villegas [RV99].) We see from this point of view that the Riemann zeta function and the L-functions 2010 Mathematics Subject Classification. Primary 11R06; Secondary 11M06, 11R42. Key words and phrases. Mahler measure; Boyd–Lawton formula; regulator. This work was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 355412-2013], the Fonds de recherche du Qu´ ebec - Nature et technologies [Projet de recherche en ´ equipe 256442], and the Institut des sciences math´ ematiques. 1