A Natural Series for the Natural Logarithm

Rodriguez Villegas expressed the Mahler measure of a polynomial in terms of an infinite series. Lück’s combinatorial L-torsion leads to similar series expressions for the Gromov norm of a knot complement. In this note we show that those formulae yield interesting power series expansions for the logarithm function. This generalizes an infinite series of Lehmer for the natural logarithm of 4. 1 The abelian case: the Mahler measure For a Laurent polynomial P in the group ring C[Z] let the conjugate P ∗ be defined by sending every g ∈ Z to g and every coefficient ag to its complex conjugate ag. The (logarithmic) Mahler measure (see e.g. [EW99]) of P is given by: