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Let me start from one example. It is known that (see J.W. Porter, On a theorem of Heilbronn) $$H(b)=\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}\ell(a/b)= \dfrac{2\log 2}{\zeta(2)}\cdot\log b+C_P-1+O_\varepsilon(b^{-1/6+\varepsilon}),$$ where $\ell(a/b)$ is a length of standart continued fraction expansion and $C_P$ is Porter's constant. Averaging over numerators and denominators one can prove asymptotic formula for the variance. Let $$E(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\ell(a/b) $$ and $${D}(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\left( \ell(a/b)-E(R)\right)^2. $$ Then (see D. Hensley, The Number of Steps in the Euclidean Algorithm) $${D}(R)=D_1\cdot\log R+o(\log R). $$ But if denominator is fixed then for the variance only right oder bound is known (see Bykovskii V.A., Estimate for dispersion of lengths of continued fractions): $$\dfrac{1}{b}\sum\limits_{a=1}^{b}\left(\ell\left(\dfrac{a}{b}\right)- \dfrac{2\log2}{\zeta(2)}\log b\right)^2\ll\log b.$$

Conjecture: $$\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}(\ell(a/b)-H(b))^2=D_1\log b+o(\log b).$$

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.

Let me start from one example. It is known that (see J.W. Porter, On a theorem of Heilbronn) $$H(b)=\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}\ell(a/b)= \dfrac{2\log 2}{\zeta(2)}\cdot\log b+C_P-1+O_\varepsilon(d^{-1/6+\varepsilon}),$$ where $\ell(c/d)$ is a length of standart continued fraction expansion and $C_P$ is Porter's constant. Averaging over numerators and denominators one can prove asymptotic formula for the variance. Let $$E(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\ell(a/b) $$ and $${D}(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\left( \ell(a/b)-E(R)\right)^2. $$ Then (see D. Hensley, The Number of Steps in the Euclidean Algorithm) $${D}(R)=D_1\cdot\log R+o(\log R). $$ But if denominator is fixed then for the variance only right oder bound is known (see Bykovskii V.A., Estimate for dispersion of lengths of continued fractions): $$\dfrac{1}{b}\sum\limits_{a=1}^{b}\left(\ell\left(\dfrac{a}{b}\right)- \dfrac{2\log2}{\zeta(2)}\log b\right)^2\ll\log b.$$

Conjecture: $$\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}(\ell(a/b)-H(b))^2=D_1\log b+o(\log b).$$

Let me start from one example. It is known that (see J.W. Porter, On a theorem of Heilbronn) $$H(b)=\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}\ell(a/b)= \dfrac{2\log 2}{\zeta(2)}\cdot\log b+C_P-1+O_\varepsilon(b^{-1/6+\varepsilon}),$$ where $\ell(a/b)$ is a length of standart continued fraction expansion and $C_P$ is Porter's constant. Averaging over numerators and denominators one can prove asymptotic formula for the variance. Let $$E(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\ell(a/b) $$ and $${D}(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le R}\sum\limits_{a\le b}\left( \ell(a/b)-E(R)\right)^2. $$ Then (see D. Hensley, The Number of Steps in the Euclidean Algorithm) $${D}(R)=D_1\cdot\log R+o(\log R). $$ But if denominator is fixed then for the variance only right oder bound is known (see Bykovskii V.A., Estimate for dispersion of lengths of continued fractions): $$\dfrac{1}{b}\sum\limits_{a=1}^{b}\left(\ell\left(\dfrac{a}{b}\right)- \dfrac{2\log2}{\zeta(2)}\log b\right)^2\ll\log b.$$

Conjecture: $$\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}(\ell(a/b)-H(b))^2=D_1\log b+o(\log b).$$