Distribution of Reduced Quadratic Irrationals

Let Φ(N) denote the number of products of matrices [ 1 1 0 1 ] and [ 1 0 1 1 ] of trace equal to N , and Ψ(N) = N n=3 Φ(n) be the number of such products of trace at most N . We prove an asymptotic formula of type Ψ(N) = c1N 2 logN + c2N 2 + Oε(N ) as N → ∞. As a result, the Dirichlet series ∞ n=3 Φ(n)n −s has a meromorphic extension in the half-plane <(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.