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The average length of reduced regular continued fractions. (English. Russian original) Zbl 1223.11010

Sb. Math. 200, No. 8, 1181-1214 (2009); translation from Mat. Sb. 200, No. 8, 79-110 (2009).
Every rational number \(a/b\) has a unique expansion as a by-excess continued fraction, i.e., can be written in the form \[ a/b=t_0- \cfrac{ 1 }{ t_1-\cfrac{ 1 }{ t_2-\cfrac{ 1 }{ \cfrac{ \ddots }{ t_l }}}}, \] with \(t_n\) integers and \(t_n\geq2\) for \(n\geq1\). Denoting by \(l(a/b)\) the length of the expansion, let \(N(R)\) be \(\sum_{a\leq b\leq R}l(a/b)\). This paper, extending previous results by B. Vallée and others, provides an asymptotic formula for the expected value \(E(R)=\bigl([R]\cdot([R]+1)\bigr)^{-1}2N(R)\). The formula has the form \[ E(R)=C_2\log^2R+C_1\log R+C_0+O(R^{-1}\log^5R), \] with \(C_0,C_1,C_2\) explicitly given constants. A similar formula is proved for the case in which only fractions \(a/b\) with relatively prime \(a,b\) are taken into consideration.

MSC:

11A55 Continued fractions
11K50 Metric theory of continued fractions
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