On Gauss-Kuz'min statistics in short intervals
The paper deals with Gauss-Kuz'min statistics. Let \([0;a_1,\dots,a_s]=\frac{a}{b}\) be the classical continued fraction, then Gauss-Kuz'min statistics are defined as NEWLINE\[NEWLINEs_{x}(\tfrac{a}{b})=\#\{j: 1\leq j\leq s, [0;a_j,\dots,a_s]\leq x\}.NEWLINE\]NEWLINE The main result of the paper is a new asymptotic formula for the following sum NEWLINE\[NEWLINE \sum_{1\leq a\leq b\atop (a,b)=1}\left(s_{\beta}(\tfrac{a}{b})-s_{\alpha}(\tfrac{a}{b})\right). NEWLINE\]NEWLINE The proof is similar to the previous author's proof of the asymptotic formula for the sum \(\sum_{1\leq a\leq b\atop (a,b)=1}s_{x}(\tfrac{a}{b}). \) It should be mentioned that classical bounds of Kloosterman sums and van der Corput method for estimating exponential sums play an important role in the proof.
scientific article; zbMATH DE number 6077460