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 √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 2x. As a corollary, they derived a new upper estimate for the number of d such that T(d)>α√x. In this paper, we improve slightly this result of three authors.
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Maxim Aleksandrovich Korolev (2024) studied this question.
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