**I. Zagier's continued fraction**


As pointed out by Gorodetsky in [his answer](https://mathoverflow.net/a/447228/12905), Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $\color{red}{c=-27,}$ then,


$$C\_2=\cfrac{1}{-3 + \cfrac{1^4\,c}{-21 + \cfrac{2^4\, c}{-57+ \cfrac{3^4\,c}{-111 +\ddots }}}}$$


or more compactly,


$$C\_2(n) = \frac1{-3 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-27k^4}{-(9k^2+9k+3)}}}$$




---


**II. Multiple Limits**


From the paper "[*Continued Fractions with Multiple Limits*](https://digitalcommons.wcupa.edu/cgi/viewcontent.cgi?article=1053&context=math_facpub)", it turns out a cfrac can have ***multiple*** limits, a famous one due to Ramanujan (but of course) with *two* limits depending on its odd or even approximants.


Zagier's cfrac $C\_2(n)$ seems to have ***six limits***, based on approximants $\text{mod}\; 6,$


\begin{align}
\lim\_{m\to\infty} C\_2(6m+0)& \overset{\color{red}?}= -0.3906\dots = -\frac{2}{3\sqrt3}\kappa\\
\lim\_{m\to\infty} C\_2(6m+1)& \overset{\color{red}?}= -0.6343\dots\\
\lim\_{m\to\infty} C\_2(6m+2)& \overset{\color{red}?}= -1.1217\dots\\
\lim\_{m\to\infty} C\_2(6m+3)& = \;\;divergent\\
\lim\_{m\to\infty} C\_2(6m+4)& \overset{\color{red}?}= +0.3404\dots\\
\lim\_{m\to\infty} C\_2(6m+5)& \overset{\color{red}?}= -0.1469\dots\\
\end{align}


The trend is more visible in the table below:


$$\begin{array}{|c|c|c|c|c|c|}
\hline
m&0&1&2&3&4&5\\
\hline
5000&-0.3906430& -0.634340& -1.121715& >10^4&0.340448& -0.146952\\
\hline
16666&-0.3906499& -0.634343& -1.121728& >10^5& 0.340435& -0.146955\\
\hline
50000&\color{blue}{-0.3906508}& -0.634344& -1.121731& >>10^5& 0.340432& -0.146956\\
\hline
\end{array}$$


Excepting $C\_2(6m+3)$, they seem to be converging to certain values as $m$ increases, though ***very slowly***. (My thanks to user **Domen** from the *Mathematica SE* for extending the table two more layers.) The only one with an apparent closed-form is the first,


$$-\frac{2}{3\sqrt3}\kappa = \color{blue}{-0.3906512}$$


where $\kappa = \operatorname{Cl}\_2\left(\tfrac13\pi\right)$ is *[Gieseking's constant](https://mathworld.wolfram.com/GiesekingsConstant.html)* (which also appears in Zagier's other cfracs.)




---


**III. Question**


1. With one obvious exception, are the rest actually converging to something?
2. If so, do they have closed-forms and is the first guess correct?




---


**P.S.** In Mathematica, the command is,



> 
> N[1/(-3 + ContinuedFractionK[-27k^4,-(9k^2+9k+3), {k, 1, n}]),20]
> 
> 
> 



On Zagier's missing continued fraction with multiple limits?
Set $Q=(1/2)L(\chi\_{-3},2)$ (related to your Gieseking constant) and
$P=2\pi^2/81$. The limits are almost certainly (not proved),


\begin{align}
\lim\_{m\to\infty}C\_2(6m+0) &= -Q\\
\lim\_{m\to\infty}C\_2(6m+1) &= -P-Q\\
\lim\_{m\to\infty}C\_2(6m+2) &= -3P - Q\\
\lim\_{m\to\infty}C\_2(6m+3) &= \infty\\
\lim\_{m\to\infty}C\_2(6m-2) &= 3P - Q\\
\lim\_{m\to\infty}C\_2(6m-1) &= P - Q
\end{align}


Remark: I use a powerful extrapolation method explained for instance in a recent book of mine with K. Belabas, and in a few seconds obtain the limits to 38 decimals, which allowed me to guess the limits.


