The first and second moments for the length of the period of the continued fraction expansion for d$\sqrt {d}$-Reference-Cited by-同舟云学术

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The first and second moments for the length of the period of the continued fraction expansion for d$\sqrt {d}$

Published:2024-07-23 Issue:4 Volume:70 Page:
ISSN:0025-5793
Container-title:Mathematika
language:en
Short-container-title:Mathematika

Author:

Battistoni Francesco¹,Grenié Loïc²,Molteni Giuseppe³^ORCID

Affiliation:

1. Dipartimento di Matematica per le Scienze Economiche Finanziarie ed Attuariali Università Cattolica Milano Italy

2. Dipartimento di Ingegneria gestionale, dell'informazione e della produzione Università di Bergamo Dalmine Italy

3. Dipartimento di Matematica Università di Milano Milano Italy

Abstract

AbstractLet be any positive and non‐square integer. We prove an upper bound for the first two moments of the length of the period of the continued fraction expansion for . This allows to improve the existing results for the large deviations of .

Funder

Istituto Nazionale di Alta Matematica "Francesco Severi"

Publisher

Wiley

Link

https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/mtk.12273

Reference11 articles.

1. The length of the period of the simple continued fraction ofd1∕2

2. Length of period simple continued fraction expansion of √d

3. The length of periodic continued fractions

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

5. The length of the period of a quadratic irrationality;Podsypanin E. V.;Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),1979

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