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A natural series for the natural logarithm. (English) Zbl 1177.11055

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 \(x \geq 4\).

MSC:

11G50 Heights
05A10 Factorials, binomial coefficients, combinatorial functions
57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 0980.11026
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