JournalsrmiVol. 23, No. 3pp. 931–952

Convergents and irrationality measures of logarithms

  • Tanguy Rivoal

    Université Grenoble I, Saint-Martin-d'Hères, France
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We prove new irrationality measures with restricted denominators of the form dνmsBm\mathrm{d}_{\lfloor\nu m\rfloor}^s B^m (where B,mN,ν>0B, m \in\mathbb{N}, \nu > 0, s{0,1}s\in\{0,1\} and dm=lcm{1,2,,m}\mathrm{d}_m=\textup{lcm}\{1,2, \ldots, m\}) for values of the logarithm at certain rational numbers r>0r>0. In particular, we show that such an irrationality measure of log(r)\log(r) is arbitrarily close to 1 provided rr is sufficiently close to 1. This implies certain results on the number of non-zero digits in the bb--ary expansion of log(r)\log(r) and on the structure of the denominators of convergents of log(r)\log(r). No simple method for calculating the latter is known. For example, we show that, given integers a,c1a,c\ge 1, for all large enough b,nb, n, the denominator qnq_n of the nn--th convergent of log(1±a/b)\log(1\pm a/b) cannot be written under the form \ddνms(bc)m\dd_{\lfloor\nu m\rfloor}^s (bc)^m: this is true for a=c=1a=c=1, b12b \ge 12 when s=0s=0, resp. b2b \ge 2 when s=1s=1 and ν=1\nu=1. Our method rests on a detailed diophantine analysis of the upper Padé table ([p/q])pq0([p/q])_{p\ge q\ge 0} of the function log(1x)\log(1-x). Finally, we remark that worse results (of this form) are currently provable for the exponential function, despite the fact that the complete Padé table ([p/q])p,q0([p/q])_{p, q\ge 0} of exp(x)\exp(x) and the convergents of exp(1/b)\exp(1/b), for b1\vert b\vert \ge 1, are well-known, for example.

Cite this article

Tanguy Rivoal, Convergents and irrationality measures of logarithms. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 931–952

DOI 10.4171/RMI/519