Mean value of sums of partial quotients of continued fractions (Q650445)

Any rational number \(x\in [0; 1)\) can be expressed as an ordinary continued fraction \[ x = \frac{1}{a_1+\frac{1}{a_2+\cdots+\frac{1}{a_n}}}\qquad (a_i \in \mathbb{N},\quad a_n\geq 2),\eqno (1) \] and as a reduced regular continued fraction \[ x =1- \frac{1}{b_1-\frac{1}{b_2-\cdots+\frac{1}{b_l}}}\qquad (b_i \in \mathbb{N},\quad b_i\geq 2).\eqno (2) \] Let \(s_1(x)\) and \(s_2(x)\) be the sums of partial quotients in~(1) with odd and even indexes respectively: \[ s_1(x)=\sum\limits_{_{\substack{ i=1\\ i\equiv 1\pmod 2}}}^{n}a_i,\qquad s_2(x)=\sum\limits_{_{\substack{ i=1\\ i\equiv 1\pmod 0}}}^{n}a_i. \] Denote by \(l(x)\) the length of reduced regular continued fraction~(2). In [Sb. Math. 200, No. 8, 1181--1214 (2009); translation from Mat. Sb. 200, No. 8, 79--110 (2009; Zbl 1223.11010)] the author considered average values \[ E(R)=\frac{2}{R(R+1)}\sum\limits_{b\leq R}\sum\limits_{a\leq b}l(a/b),\qquad E^*(R)=\frac{2}{R(R+1)}\sum\limits_{b\leq R}\sum\limits_{_{\substack{ a\leq b\\ (a,b)=1}}}l(a/b). \] It was proved that \[ E(R)=c_2\log^2R+c_1\log R+c_0+O(R^{-1}\log^5R), \] \[ E^*(R)=c_2\log^2R+c_1'\log R+c_0'+O(R^{-1}\log^5R) \] with some explicit constants \(c_2\), \(c_1\), \(c_0\), \(c_1'\) and \(c_0'\). The paper under review devoted to the average values of sums of odd and even partial quotients \[ S_i(R)=\frac{2}{R(R+1)}\sum\limits_{b\leq R}\sum\limits_{a\leq b}s_1(a/b),\quad S_i^*(R)=\frac{2}{R(R+1)}\sum\limits_{b\leq R}\sum\limits_{_{\substack{ a\leq b\\ (a,b)=1}}}s_1(a/b)\quad(i=1,2). \] The author proves asymptotic formulae \[ S_1(R)=E(R)+\frac{1}{2}+O(R^{-1}),\qquad S_2(R)=E(R)-\frac{1}{2}+O(R^{-1}), \] \[ S_1^*(R)=E^*(R)+\frac{1}{2}+O(R^{-1}\log R),\qquad S_2^*(R)=E^*(R)-\frac{1}{2}+O(R^{-1}\log R). \] They lead to asymptotic formulae for \(S_i(R)\) and \(S_i^*(R)\) which seemed to be optimal up to logarithmic factor, see \textit{D. A. Frolenkov} [Sb. Math. 203, No. 2, 288--305 (2012); translation from Mat. Sb. 203, No. 2, 143--160 (2012; Zbl 1292.11139)].