Dmitry Frolenkov (Steklov Mathematical Institute of Russian Academy of Sciences): The average length of continued fractions with fixed denominator

Abstract. In 1968 H. Heilbronn proved an asymptotic formula for the normalized mean value of the length of continued fractions with fixed denominator. This result was improved by J.W. Porter in 1975. He followed the approach of Heilbronn and used Weil's bound for Kloosterman sums in order to obtain a power saving error in the asymptotic formula. In 2008 the exponent in the error term was further improved by A.V. Ustinov. We will briefly discuss all these results and how the problem under consideration is related to the additive divisor problem. This relation is one of the new key ideas of our joint work with V.A. Bykovskii.