The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$

Abstract

Let $d$ be any positive and non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$. This allows to improve the existing results for the large deviations of $T(d)$.

Some misprints have been corrected. The new version of the paper is scheduled to appear in Mathematika