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arXiv · 2403.08616

An upper bound for the second moment of the length of the period of the continued fraction expansion for $\sqrt{d}$

M. A. Korolev
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Abstract

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.Grenié and G.Molteni established (in particular) an upper bound for the second moment of $T(d)$ over the segment $x α\sqrt{x}$. In this paper, we improve slightly this result of three authors. In this version, we improve the estimate of the remainder term in the main asymptotic formula, correct some misprints and add an important remark made by F.Battistoni, L.Grenié and G.Molteni. This remark concerns the explicit value of the constant in the main term of asymptotics.

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M. A. Korolev. 2024-08-21. An upper bound for the second moment of the length of the period of the continued fraction expansion for $\sqrt{d}$. https://arxiv.org/abs/2403.08616

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