A note on the rational approximations to \(\tanh {1\over k}\) (Q1320539)

In Number theory and combinatorics, Japan 1984, 353-367 (1985; Zbl 0614.10030) \textit{I. Shiokawa} proved the following theorem. Let \(k\) be a positive integer. Then there is a constant \(C= C(k)>0\) such that \[ |\tanh (1/k)- p/q|> C(\log\log q)/( q^ 2\log q) \] for all fractions \(p/q\) \((q\geq 3)\). In the present note the author shows how to compute such constants \(C\) explicitly. The proof is based on the known continued fraction expansion of \(\tanh (1/k)\).