THE LARGEST DIGIT IN THE CONTINUED-FRACTION EXPANSION OF A RATIONAL NUMBER

The finite continued fraction sequence of a reduced fraction a/b, with 0 less-than-or-equal-to a < b, is the sequence d = (d(1), d(2), ..., d(r)) of positive integers such that d(r) > 1, and a/b = 1/(d(1) + 1/(d(2) + ... + 1/d(r))). 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 greater-than-or-equal-to 1 digits (also known as partial quotients) d(j) in this expansion (1 less-than-or-equal-to j less-than-or-equal-to 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) less-than-or-equal-to alpha log b] almost-equal-to exp(- 12/alpha-pi-2). More precisely, it is shown that for all epsilon > 0, and uniformly in alpha > epsilon as x --> infinity, #{(a, b):0 less-than-or-equal-to a < b less-than-or-equal-to x, gcd(a, b) = 1, and D(a, b) less-than-or-equal-to alpha log x} almost-equal-to (3/pi-2)x2 exp(- 12/alpha-pi-2). The question of how often there are exactly M digits exceeding alpha log b in the continued fraction expansion of a reduced fraction a/b with 0 less-than-or-equal-to a < b less-than-or-equal-to x is also touched on. Evidence points to the estimate (3/pi-2)x2(M!)-1(12/alpha-pi-2)M exp(- 12/alpha-pi-2) for the number of such fractions.