Mahler's Measure and the Dilogarithm (II)
Abstract
We continue to investigate the relation between the Mahler measure of certain two variable polynomials, the values of the BlochWigner dilogarithm $D(z)$ and the values $\zeta_F(2)$ of zeta functions of number fields. Specifically, we define a class $\A$ of polynomials $A$ with the property that $\pi m(A)$ is a linear combination of values $D$ at algebraic arguments. For many polynomials in this class the corresponding argument of $D$ is in the Bloch group, which leads to formulas expressing $\pi m(A)$ as a linear combination with unspecified rational coefficients of $V_F$ for certain number fields $F$ ($V_F := c_F\zeta_F(2)$ with $c_F>0$ an explicit simple constant). The class $\A$ contains the $A$polynomials of cusped hyperbolic manifolds. The connection with hyperbolic geometry often provides means to prove identities of the form $\pi m(A)= r V_F$ with an explicit value of $r\in \Q^*$. We give one such example in detail in the body of the paper and in the appendix.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2003
 arXiv:
 arXiv:math/0308041
 Bibcode:
 2003math......8041B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Geometric Topology;
 11C08;
 11G10;
 11G55
 EPrint:
 37 pages. Main text by Boyd and RodriguezVillegas