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arXiv · 2401.13118

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
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Abstract

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

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Francesco Battistoni, Loïc Grenié, Giuseppe Molteni. 2024-01-23. The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$. https://doi.org/10.1112/mtk.12273

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