Products of matrices $[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}]$ and $[ \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} ]$ and the distribution of reduced quadratic irrationals

Let $Φ(N)$ denote the number of products of matrices $[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}]$ and $[ \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} ]$ of trace equal to $N$, and $Ψ(N)=\sum_{n=3}^N Φ(n)$ be the number of such products of trace between $3$ and $N$. We prove an asymptotic formula of type $Ψ(N) = c_1 N^2 \log N +c_2 N^2 + O_\varepsilon (N^{7/4+\varepsilon})$ as $N\to \infty$. As a result, the Dirichlet series $\sum_{n=1}^\infty Φ(n) n^{-s}$ has a meromorphic extension in the half-plane $\Re (s)>7/4$ with a single, order two pole at $s=2$. Our estimate also improves on an asymptotic result of Faivre concerning the distribution of reduced quadratic irrationals, providing an explicit upper bound for the error term.

No reviews yet.

No comments yet.