  If $d$ is not a perfect square, we define $T(d)$ as the length of the minimal
period of the simple continued fraction expansion for $\sqrt{d}$. Otherwise, we
put $T(d) = 0$. In the recent paper (2024), F.Battistoni, L.Greni\'{e} and
G.Molteni established (in particular) an upper bound for the second moment of
$T(d)$ over the segment $x<d\leqslant 2x$. As a corollary, they derived a new
upper estimate for the number of $d$ such that $T(d)>\alpha\sqrt{x}$. In this
paper, we improve slightly this result of three authors.
