To each irrational number x is associated an infinite sequence of rational fractions (p_n/q_n), known as the convergents of x. Consider the functions q_n |q_nx-p_n |=θ_n(x). We shall primarily be concerned with the computation, for almost all real x, of the ergodic sum lim_n→∞ (1/n)∑_k=1ⁿlogθ_k(x)= -1-(1/2)log 2≈ -1.34657.