On the rational approximations to \(\tanh\frac1k\) (Q1903373)

[This is the full version of a note announced in Proc. Japan Acad., Ser. A 69, 161-163 (1993; Zbl 0806.11032) and Ser. A 70, 315-316 (1993) reviewed above.] The main result of this paper is that for every \(\varepsilon>0\) there are infinitely many solutions of the inequality \[ |\tanh (1/k)- p/q|< (1/ (2k)+ \varepsilon)\log \log q/( q^2\log q) \] in integers \(p\) and \(q\). Further, there exists a number \(Q= Q(k, \varepsilon)\) such that \[ |\tanh (1/k)- p/q |> (1/( 2k)- \varepsilon) \log \log q/( q^2 \log q) \] holds for all integers \(p\) and \(q\) with \(q>Q\). The proofs use continued fractions expansions of \(\tanh 1/k\).