TITLE: Average number of iterations for the Euclidean algorithm to terminate
QUESTION [7 upvotes]: Let $N$ be a positive integer and $0 \leq s < N$.
We try to divide $s$ into $N$ using the Euclidean algorithm:
$N = q_1 s + r_1 $
$r = q_2 r_1 + r_2 $
$\vdots$
$r_{K-1} = q_{K-1} r_K$
If we choose $-r_{i-1}/2 \leq r_i < r_{i-1}/2$, I think this determines the $q_i$'s uniquely, but I don't think this matters for my question. 
For a fixed $N$ and $0 \leq s < N$, define $K_s$ to be the number of iterations before the Euclidean algorithm terminates, (i.e. $K$ in the instance written above).
Are there existing tools to characterize
$$ \frac{1}{N} \sum_{0 \leq s < N} K_s, $$
for a given $N$?

REPLY [2 votes]: The complexity analysis of the Euclidean algorithm is much much older that the given reference. It goes back to at least Lamé, before 1785. The above answer should be viewed as modern version of Lamé's theorem. The average is actually a normal variable, there are several difficult proofs, see Morris - A short proof that the number of division steps in the Euclidean algorithm is normally distributed, and earlier references.