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

Francesco Battistoni, Loïc Grenié, Giuseppe Molteni

Published 2024-01-23Version 1

Let $d$ be any positive 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 $T(d)$.