This is a joint work with Loic Grenié and Giuseppe Molteni. Given a positive integer \(d\) which is not a square, denote by \(T(d)\) the length of the period of the continued fraction expansion for \(\sqrt{d}\). We prove upper bounds for the first and the second moment of \(T(d)\) by studying a different function \(g(d)\), originally introduced by Hickerson: we detect the asymptotic of the first moment of \(g(d)\) and an upper bound for the second moment. The results allow to improve the estimates for the size of the sets of integers \(d\) for which \(T(d) > \alpha\sqrt{d}\), with \(\alpha\) a real parameter. We also report recent progress by Korolev on the asymptotic for the second moment of \(g(d)\).