  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.
