# Convergents and irrationality measures of logarithms

### Tanguy Rivoal

Université Grenoble I, Saint-Martin-d'Hères, France

## Abstract

We prove new irrationality measures with restricted denominators of the form $d_{⌊νm⌋}B_{m}$ (where $B,m∈N,ν>0$, $s∈{0,1}$ and $d_{m}=lcm{1,2,…,m}$) for values of the logarithm at certain rational numbers $r>0$. In particular, we show that such an irrationality measure of $g(r)$ is arbitrarily close to 1 provided $r$ is sufficiently close to 1. This implies certain results on the number of non-zero digits in the $b$--ary expansion of $g(r)$ and on the structure of the denominators of convergents of $g(r)$. No simple method for calculating the latter is known. For example, we show that, given integers $a,c≥1$, for all large enough $b,n$, the denominator $q_{n}$ of the $n$--th convergent of $g(1±a/b)$ cannot be written under the form \( \dd_{\lfloor\nu m\rfloor}^s (bc)^m \): this is true for $a=c=1$, $b≥12$ when $s=0$, resp. $b≥2$ when $s=1$ and $ν=1$. Our method rests on a detailed diophantine analysis of the upper Padé table $([p/q])_{p≥q≥0}$ of the function $g(1−x)$. Finally, we remark that worse results (of this form) are currently provable for the exponential function, despite the fact that the complete Padé table $([p/q])_{p,q≥0}$ of $exp(x)$ and the convergents of $exp(1/b)$, for $∣b∣≥1$, 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