HIGHER MAHLER MEASURE OF AN n-VARIABLE FAMILY MATILDE N. LAL ´ IN AND JEAN-S ´ EBASTIEN LECHASSEUR Abstract. We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at cer- tain roots of unity. 1. Introduction For k a positive integer, the k-higher Mahler measure of a non-zero, n-variable, rational function P (x 1 ,...,x n ) ∈ C(x 1 ,...,x n ) is given by m k (P (x 1 ,...,x n )) =  1 0 ...  1 0 log k   P ( e 2πiθ 1 ,...,e 2πiθn )  dθ 1 ...dθ n . We observe that the case k = 1 recovers the formula for the “classical” Mahler measure. This function, originally defined as a height on polynomi- als, has attracted considerable interest in the last decades due to its con- nection to special values of the Riemann zeta function, and of L-functions associated to objects of arithmetic significance such as elliptic curves as well as special values of polylogarithms and other special functions. Part of such phenomena has been explained in terms of Beilinson’s conjectures via rela- tionships with regulators by Deninger [Den97] (see also the crutial articles by Boyd [Boy97] and Rodriguez-Villegas [R-V97]). Higher (and multiple) Mahler measures were originally defined in [KLO08] and subsequently studied by several authors [Sas10, BS11, LS11, BBSW12, BS12, Sas12, Bis14, BM14]. A related object, the Zeta Mahler measure, was first studied by Akatsuka in [Aka09]. As remarked by Deninger, higher Mahler measures are expected to yield different regulators than the ones that appear in the case of the usual Mahler measure and they may reveal a more complicated structure at the level of the periods (see [Lal10] for more details). In order to continue this program of understanding periods that arise from higher Mahler measure, an essential component is to generate examples 2010 Mathematics Subject Classification. Primary 11R06; Secondary 11M06, 11R09. Key words and phrases. Mahler measure, Higher Mahler measure, special values of ζ (s) and Dirichlet L-functions, polylogarithms. 1