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Mat. Sb., 2017, Volume 208, Number 5, Pages 63–102 (Mi msb8718)  

The average length of finite continued fractions with fixed denominator

V. A. Bykovskiia, D. A. Frolenkovba

a Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: In 1968 Heilbronn proved an asymptotic formula for the mean value of the lengths of continued fraction expansions of rational numbers with identical denominators. A new method is proposed for solving Heilbronn's problem and its generalizations. New estimates for the remainders, which improve the earlier results due to Porter (1975) and Ustinov (2005), are obtained.
Bibliography: 28 titles.

Keywords: continued fraction, additive divisor problem, convolution formula.

Funding Agency Grant Number
Russian Science Foundation 14-11-00335
This research was financed by a grant from the Russian Science Foundation (project no. 14-11-00335) at the Khabarovsk Division of the Institute of Applied Mathematics of the Far-Eastern Branch of the Russian Academy of Sciences.


DOI: https://doi.org/10.4213/sm8718

Full text: PDF file (805 kB)
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English version:
Sbornik: Mathematics, 2017, 208:5, 644–683

Bibliographic databases:

Document Type: Article
UDC: 517.524+511.335+517.58
MSC: 11K50
Received: 13.04.2016 and 30.11.2016

Citation: V. A. Bykovskii, D. A. Frolenkov, “The average length of finite continued fractions with fixed denominator”, Mat. Sb., 208:5 (2017), 63–102; Sb. Math., 208:5 (2017), 644–683

Citation in format AMSBIB
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