We prove new irrationality measures with restricted denominators of the form (where , and ) for values of the logarithm at certain rational numbers . In particular, we show that such an irrationality measure of is arbitrarily close to 1 provided is sufficiently close to 1. This implies certain results on the number of non-zero digits in the --ary expansion of and on the structure of the denominators of convergents of . No simple method for calculating the latter is known. For example, we show that, given integers , for all large enough , the denominator of the --th convergent of cannot be written under the form : this is true for , when , resp. when and . Our method rests on a detailed diophantine analysis of the upper Padé table of the function . Finally, we remark that worse results (of this form) are currently provable for the exponential function, despite the fact that the complete Padé table of and the convergents of , for , are well-known, for example.
Cite this article
Tanguy Rivoal, Convergents and irrationality measures of logarithms. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 931–952DOI 10.4171/RMI/519