This article is cited in 2 papers

The average length of finite continued fractions with fixed denominator

V. A. Bykovskii^a, D. A. Frolenkov<sup>ba

a</sup> 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.

UDC: 517.524+511.335+517.58

MSC: 11K50

Received: 13.04.2016 and 30.11.2016

DOI: 10.4213/sm8718

 English version: Sbornik: Mathematics, 2017, 208:5, 644–683

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