The mean number of steps in the Euclidean algorithm with odd partial quotients (Q650319)

The paper deals with different kinds of the Euclidean algorithm and corresponding continued fractions. The main result is a new asymptotic formula for the mean value of steps \(h(\frac{a}{b})\) in the Euclidean algorithm with odd partial quotients. It should be mentioned that asymptotic formulas are obtained both in the case of averaging over numerators and in the case of averaging over both numerators and denominators. These asymptotic formulas improve the previous result due to Baladi and Vallée. The main idea of the proof is to express the value \(h(\frac{a}{b})\) in terms of Gauss-Kuzmin statistics. And for this statistics asymptotic formulas with the desired error terms had been previously obtained by the author.