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I. Zagier's continued fraction

As pointed out by Gorodetsky in his answer, 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", 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 (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]

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2 Answers 2

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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.

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    $\begingroup$ Beautiful! Gieseking and $\zeta(2)$ together. Do you how many minutes it took Wolfram Alpha to find just 4 decimals? 😊 $\endgroup$ May 22, 2023 at 18:46
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    $\begingroup$ By the way, i already checked the 2 remaining cfracs I marked with “??” in my other post, and it was hard to find a pattern to the approximants mod 6 or mod 8, if there is any at all. $\endgroup$ May 22, 2023 at 18:53
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    $\begingroup$ There are 6 beautiful cfracs with $L(\chi_{-3},2)$ and $\zeta(2)$ together, 4 of them are given in my paper arXiv:2212.01095. Interestingly enough, these 4 are exactly the linear combinations in my answer above. $\endgroup$ May 22, 2023 at 20:46
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    $\begingroup$ Reading that paper right now. By the way, I made some minor variable changes to your answer to make it look more symmetrical and aesthetic. I hope it's ok. $\endgroup$ May 23, 2023 at 9:13
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To complete the 12 cfracs in this post and the 4 in the next, all associated with 16 "sporadic sequences", then 13 of them have closed-forms, 1 has six limits (also with closed-forms but one divergent), and the last 2 as divergent.

We evaluate the last 3 cfracs for $n = 1000\; \text{to}\; 1200,$ sort the values $v$, and plot the values $-2<v<2.$


I. Degree 2 for (-9,-3,-27)

$$C_2(-9,-3,-27) = \frac1{-3 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-27k^4}{(-9k^2-9k-3)}}}$$

As discussed in the post above, this has six limits as clearly seen in the plot below (the divergent $v \to \infty$ is not shown),

Cfrac27


II. Degree 3 for (11,3,1)

$$C_3(11,3,1) = \frac1{-5 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-125k^4}{-(2k+1)(11k^2+11k+5)}}}$$

Its plot is vastly different and presumably has infinitely many limits within $-1<v<0$,

Cfrac125

It has an illusory "pattern" if $n$ is mod $5$ or mod $7$ which misled me for a while, but it disappeared with further analysis.


III. Degree 3 for (7,2,8)

$$C_3(7,2,8) = \frac1{-3 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-81k^4}{-(2k+1)(7k^2+7k+3)}}}$$

Likewise, its plot looks very similar to the previous one,

Cfrac81b

so the same conclusion about it can be made.

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