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

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

2005-03-09

arxiv.org/abs/math/0503186v3

Mathematics

Number Theory

16 pages

Scientific paper

Let $\Phi(N)$ denote the number of products of matrices $[ 1 & 1 // 0 & 1 ]$ and $[ 1 & 0 // 1 & 1 ]$ of trace equal to $N$, and $\Psi(N)=\sum_{n=1}^N \Phi(n)$ be the number of such products of trace at most $N$. We prove an asymptotic formula of type $\Psi(N)=c_1 N^2 \log N+c_2N^2 +O_\eps(N^{7/4+\eps})$ as $N\to \infty$. As a result, the Dirichlet series $\sum_{n=1}^\infty \Phi(n) n^{-s}$ 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.

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