Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms (Q2884667)

For \(a/b\in(0,1]\) denote by \(s(a/b)\) the length of the standard continued fraction expansion NEWLINE\[NEWLINE\frac{a}{b}=\frac{1}{a_1+{\atop\ddots\,\displaystyle{+\frac{1}{a_s}}}},NEWLINE\]NEWLINE and let \(l(a/b)\) be a length of reduced regular continued fraction NEWLINE\[NEWLINE\frac{a}{b}=1-\frac{1}{b_1-{\atop\ddots\,\displaystyle{-\frac{1}{b_l}}}}\qquad(b_1, \ldots, b_l\geq 2).NEWLINE\]NEWLINE It was known (see [\textit{A. V. Ustinov}, Izv. Math. 72, No. 5, 1023--1059 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 5, 189--224 (2008; Zbl 1211.11009)]; [\textit{E. N. Zhabitskaya}, Math. Notes 89, No. 3, 450--454 (2011); translation from Mat. Zametki 89, No. 3, 472--472 (2011; Zbl 1306.11010)]) that average lengths NEWLINE\[NEWLINEE^+(R)=\frac{2}{[R]([R]+1)}\sum\limits_{b\leq R}\sum\limits_{a\leq b}s\left(\frac{a}{b}\right),\qquad E^-(R)=\frac{2}{[R]([R]+1)}\sum\limits_{b\leq R}\sum\limits_{a\leq b}l\left(\frac{a}{b}\right)NEWLINE\]NEWLINE satisfy asymptotic formulae NEWLINE\[NEWLINEE^+(R)=c_1^+\log R+c_0^++O(\omega^+(R)),\qquad E^-(R)=c_2^+\log^2 R+c_1^+\log R+c_0^++O(\omega^-(R))NEWLINE\]NEWLINE with some explicit constants \(c_i^\pm\) and the error terms NEWLINE\[NEWLINE\omega^\pm(R)=O(R^{-1}\log^5R).NEWLINE\]NEWLINE The author proves sharper estimates NEWLINE\[NEWLINE\omega^+(R)=O(R^{-1}\log^3R),\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\omega^-(R)=O(R^{-1}\log^3R). \tag{2}NEWLINE\]NEWLINE It is interesting that the proof of the estimate (2) uses some ideas from Selberg's elementary proof of the prime number theorem.NEWLINENEWLINEEstimates (1) and (2) are nearly optimal, although \(\Omega\)-theorems for \(\omega^\pm(R)\) are not known. According to numerical experiments performed by the author, the error terms \(\omega^\pm(R)\) should satisfy the estimates NEWLINE\[NEWLINE\omega^+(R)=O(R^{-1}\log R),\qquad \omega^-(R)=O(R^{-1}\log^2 R).NEWLINE\]