The average length of Minkowski's continued fractions (Q2906404)

Let \(R\) be real positive number and \(c\) and \(d\) \((c<d)\) be natural numbers. The author of this paper discusses the following sum: NEWLINE\[NEWLINEE'(R)=\frac{2}{[R]([R]+1)}\sum_{d\leq R}\sum_{c\leq d}s(c/d;1),NEWLINE\]NEWLINE where \(s(c/d;1)\) means the length of the Minkowski's diagonal continued fraction expansion of \(c/d\) with parameter \(\Omega = 1\); \([R]\) means the integer part of \(R\). She proves the following Theorem. NEWLINE\[NEWLINEE'(R) =\frac{\lg R}{\zeta(2)}+C_{d}+O(R^{-1}\lg^{3}R),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEC_{d}= \frac{1}{\zeta(2)}\left(2\gamma-\frac{3}{2}+2\lg\left(\frac{3}{2}\right)(1-\lg 2)-\lg^{2}3-2\, \text{Li}_{2}\left(\frac{2}{3}\right)+2\, \text{Li}_{2}\left(\frac{-1}{2}\right)-\frac{\zeta'(2)}{\zeta(2)}\right)-\frac{17}{6}.NEWLINE\]NEWLINE The proof is based on 8 auxiliary lemmas.