Estimates of the Levy constant for \(\sqrt{p}\) and class number one criterion for \(\mathbb{Q}(\sqrt{p})\) (Q1807477)

Let \(p\) be a prime congruent to \(3\pmod 4\), let \(\ell(\sqrt{p})\) be the length of the period of the continued fraction of \(\sqrt{p}\) and let \(h_K\) be the class number of \(K= \mathbb{Q}(\sqrt{p})\). For \(p>91\) the author gives a necessary and sufficient condition for the class number of \(K\) to be \(1:h_K=1\) if and only if \(\ell(\sqrt{p})>0,56\sqrt{p} L(1,\chi)\) where \(L(1,\chi)\) is the \(L\)-series attached to the unique quadratic character \(\chi\bmod p\).