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The finite continued
fraction sequence of a reduced fraction a∕b, with 0 ≤ a < b, is the sequence
d = (d(1),d(2),…,d(r)) of positive integers such that d(r) > 1, and
In the standard terminology of continued fractions, this is written as
[0;d(1),d(2),…,d(r)], which we abbreviate here to [d(1),d(2),…,d(r)]. Thus
[1,4,2] = 1∕(1 + 1∕(4 + 1∕2)) = 9∕11. The empty sequence corresponds to 0∕1.
For any other fraction, there will be r ≥ 1 digits (also known as partial
quotients) d(j) in this expansion (1 ≤ j ≤ r). The largest of these we call
D(a∕b) or D(a,b). Thus D(9∕11) = D(9,11) = 4. The aim of this work is to
elucidate the distribution of D(a,b). Put informally, the main result is that
Prob[D(a,b) ≤ α log b] ≈ exp(−12∕απ2). More precisely, it is shown that for all 𝜀 > 0,
and uniformly in a α > 𝜀 as x →∞,
The question of how often there are exactly M digits exceeding α log b in the
continued fraction expansion of a reduced fraction a∕b with 0 ≤ a < b ≤ x is also
touched on. Evidence points to the estimate
for the number of such fractions.
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