The Number of Steps in the Euclidean Algorithm

Abstract

AbstractWhen the Euclidean algorithm is run on a pair (a, b) of positive integers with a < b, the number of steps of the form (a, b) → (b mod a, a) required to extract the gcd is known to be typically close to (12 log 2/π2) log b. Here we sharpen earlier estimates of Dixon [J. Number Theory2 (1970), 414-422] and Heilbronn ["Number Theory and Analysis (Papers in Honor of Edmund Landau)," pp. 87-96] and show that among pairs (a, b) with 1 ≤ a ≤ b ≤ x as x → ∞, the number of such steps has an asymptotically normal distribution with mean close to (12 log 2/π2) log x and variance close to C1 log x, where C1 = π−6(12 log 2/π2)3(d2/ds2) log λ(s)|s = 2, with λ(s) denoting a certain pseudo zeta function associated with continued fractions