Number Theory – Zagier’s Missing Continued Fraction with Multiple Limits

closed-form-expressionscontinued-fractionslimits-and-convergencent.number-theory

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]

Best Answer

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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