Products of matrices [1 0 11] and the distribution of reduced quadratic irrationals

Let (N) denote the number of products of matrices and of trace equal to N, and be the number of such products of trace at most N. We prove an asymptotic formula of type as N → ∞. As a result, the Dirichlet series has a meromorphic extension in the half-plane 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.