On the period of the continued fraction expansion for $\sqrt{d}$
- 作者: Korolev M.A.1
-
隶属关系:
- Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
- 期: 卷 89, 编号 1 (2025)
- 页面: 30-53
- 栏目: Articles
- URL: https://bakhtiniada.ru/1607-0046/article/view/303936
- DOI: https://doi.org/10.4213/im9587
- ID: 303936
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详细
If $d$ is not a perfect square, we define $T(d)$ as the length of theminimal period of the simple continued fraction expansion for $\sqrt{d}$.Otherwise, we put $T(d)=0$. In the recent paper (2024), F. Battistoni,L. Grenie and G. Molteni established (in particular) an upper boundfor the second moment of $T(d)$ over the segment $x
作者简介
Maxim Korolev
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
编辑信件的主要联系方式.
Email: korolevma@mi-ras.ru
Doctor of physico-mathematical sciences, no status
参考
- D. R. Hickerson, “Length of period simple continued fraction expansion of $sqrt{d}$”, Pacific J. Math., 46:2 (1973), 429–432
- F. Battistoni, L. Grenie, G. Molteni, “The first and second moment for the length of the period of the continued fraction expansion for $sqrt{d}$”, Mathematika, 70:4 (2024), e12273, 12 pp.
- A. M. Rockett, P. Szüsz, “On the lengths of the periods of the continued fractions of square-roots of integers”, Forum Math., 2:2 (1990), 119–123
- C. Hooley, “On the number of divisors of quadratic polynomials”, Acta Math., 110 (1963), 97–114
- D. I. Tolev, “On the exponential sum with square-free numbers”, Bull. London Math. Soc., 37:6 (2005), 827–834
- Н. М. Коробов, Тригонометрические суммы и их приложения, Наука, М., 1989, 240 с.
- S. Bettin, V. Chandee, “Trilinear forms with Kloosterman fractions”, Adv. Math., 328 (2018), 1234–1262
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