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
Boca, Florin P.
arXiv, 0503186 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2005-03-09
Classification:  Mathematics - Number Theory,  Mathematical Physics,  11N37,  11Z05,  82B20,  82B26 
@article{0503186,
     author = {Boca, Florin P.},
     title = {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},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0503186}
}

Boca, Florin P. 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. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0503186/