The range of multiplicative functions on ℂ[ x ], ℝ[ x ] and ℤ[ x ]

Mahler's measure is generalized to create the class of multiplicative distance functions . These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following the work of Chern and Vaaler ( J. Reine Angew. Math. ), 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 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 2 N has an analytic continuation to rational functions with rational coefficients, simple poles at integers between− N and N , and a zero of multiplicity 2 N 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 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.