  Mahler's measure is generalized to create the class of {\it multiplicative
distance functions}. These functions measure the complexity of polynomials
based on the location of their zeros in the complex plane. Following work of
S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each
multiplicative distance function two families of analytic functions which
encode information about its range on \C[x] and \R[x]. These {\it moment
functions} are Mellin transforms of distribution functions associated to the
multiplicative distance function and demonstrate a great deal of arithmetic
structure. For instance, we show that the moment function associated to
Mahler's measure restricted to real reciprocal polynomials of degree 2N has an
analytic continuation to rational functions with rational coefficients, simple
poles at integers between -N and N, and a zero of multiplicity 2N at the
origin. This discovery leads to asymptotic estimates for the number of
reciprocal integer polynomials of fixed degree with Mahler measure less than T
as $T \to \infty$. To explain the structure of this moment functions we show
that the real moment functions of a multiplicative distance function can be
written as Pfaffians of antisymmetric matrices formed from a skew-symmetric
bilinear form associated to the multiplicative distance function.
