Calculation of the variance in a problem in the theory of continued fractions

We study the random variable N(α,R)= ++{j≥1:Qj(α)≤R}, where α element of [0;1) and Pj(α)/Qj(α) is the jth convergent of the continued fraction expansion of the number α=[0;t1,t2,...]. For the mean value N(R)=∫01N(α,R) dα and variance D(R)=∫01(N(α,R)-N(R))2dα of the random variable N(α,R), we prove the asymptotic formulae with two significant terms N(R)=N1log R+N0+O(R-1+ε), D(R)=D1log R+D0+O(R11/3+ε. Bibliography: 13 titles.