Higher Mahler measures and zeta functions

Abstract

We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of ${\log }^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log |P|$. A zeta Mahler measure, involving the integral of $|P{|}^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.