High moments of the Estermann function

For a/q ∈ ℚ the Estermann function is defined as and by meromorphic continuation otherwise. For q prime, we compute the moments of D(s, a/q) at the central point s = 1/2, when averaging over 1 ≤ a < q. As a consequence we deduce the asymptotic for the iterated moment of Dirichlet L-functions obtaining a power saving error term. Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing is the continued fraction expansion of a/q we prove that for k ≥ 2 and q primes one has as → ∞.