Dasbach, Oliver T. A natural series for the natural logarithm. (English) Zbl 1177.11055 Electron. J. Comb. 15, No. 1, Research Paper N5, 7 p. (2008). The paper brings together and compares two related results. The first is a series due to Rodriguez-Villegas for the Mahler measure of a polynomial in several variables. The second is a series due to Lück for the volume of a hyperbolic knot complement. By applying the formula of Rodriguez-Villegas to the cyclotomic polynomials \(1+a+\dots+a^m\) which has Mahler measure equal to \(0\), he obtains some interesting formulas for the natural logarithm. For example, if \(m = 1\), he obtains \(\ln x = \sum_{n \geq 1} \sum_{j=0}^n {1 \over n} {n\choose j} {2j\choose j} ({-1 \over x})^j\), valid for \(x \geq 4\). Reviewer: David W. Boyd (Vancouver) MSC: 11G50 Heights 05A10 Factorials, binomial coefficients, combinatorial functions 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:Mahler measure; combinatorial \(L^2\)-torsion; knot complement; volume; Gromov norm Citations:Zbl 0980.11026 PDFBibTeX XMLCite \textit{O. T. Dasbach}, Electron. J. Comb. 15, No. 1, Research Paper N5, 7 p. (2008; Zbl 1177.11055) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.