The largest digit in the continued fraction expansion of a rational number (Q1183092)

Let \(D(a,b)\) be the largest partial quotient in the continued fraction of the rational \(a/b\), \(0\leq a<b\leq x\), gcd\((a,b)=1\). The author considers \(\Phi(x,\alpha)=\#\{(a,b): D(a,b)\leq\alpha\log x\}\) and proves (1) \(\Phi(x,\alpha)\sim(3/\pi^ 2)x^ 2\exp(-12/\alpha\pi^ 2)\) uniformly in \(\alpha\geq 4/\log\log x\) as \(x\to\infty\). The basic idea of the proof is to determine \(\Phi(x,\alpha)\) from the number of fractions with at least \(j\) digits too large. More general, an asymptotic estimate for the number of fractions with exactly \(M\) digits \(\geq\alpha\log x\) is obtained. An estimate corresponding to (1) holds also for \(F(x,\alpha)=\#\{(a,b): \delta(a,b)>1/\alpha\log x\}\), where \(\delta(a,b)=\min_{1\leq k<b}\| ka/b\|\). The results are somewhat related to those by \textit{J. Galambos} [Q. J. Math., Oxf., II. Ser. 23, 147-151 (1972; Zbl 0234.10041)].