Geodesic
On the period of the continued fraction expansion for d
Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 26-49.

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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. Grenié and G. Molteni established (in particular) an upper bound for the second moment of T(d) over the segment x. As a corollary, they derived a new upper estimate for the number of d such that T(d)>αx. In this paper, we slightly improve this result of three authors.
Mots-clés : continued fraction, period of simple continued fraction expansion, trilinear Kloosterman sums. 
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M. A. Korolev. On the period of the continued fraction expansion for $\sqrt{d}$. Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 26-49. https://geodesic-test.mathdoc.fr/item/IM2_2025_89_1_a2/

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