Proceedings of the London Mathematical Society Advance Access originally published online on December 18, 2007
Proceedings of the London Mathematical Society 2008 96(3):697-737; doi:10.1112/plms/pdm037
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© 2007 London Mathematical Society
The range of multiplicative functions on 
[x], 
[x] and 
[x]
Current address:
Department of Mathematics
University of Colorado at Boulder
Boulder, CO 80309-0395
USA
christopher.sinclair@colorado.edu
Received 23 January 2007. Revision received 24 April 2007.
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 
[x] and 
[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 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 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.
2000 Mathematics Subject Classification 11B83 (primary), 11J71, 37A45, 60G10 (secondary).