Ramanujan's $\tau(n)$ and continued fractions

Question

In D.H. Lehmer's paper "Ramanujan's function $\tau(n)$, (Duke J. Math v. 10 1943, pp. 483-492), Lehmer states the Ramanujan conjecture $|\tau( p )|< 2p^{11/2}$, so that $p^{-11/2}\tau( p )=2\cos(\theta_p)$ where $\theta_p$ is real. He observes "It is interesting to note that $2\cos(\theta_{11})=1.000872909\ldots$" ie that $\theta_{11}$ is very nearly $\pi/3$.

In fact, Mathematica computes that $\theta_{11}/\pi=0.333172889904775\ldots$ There's nothing special about the expansion as a decimal, so I looked instead at the continued fraction expansion, which Mathematica can also do. One finds it is $ \{0, 3, 692, 5, 4, 1, 2, 3, 1, 2,\ldots\} $ The $3$ is expected, as is the fact that the next term is large. But that it's 692 is surprising to me, recalling that $\tau(n)\equiv \sigma_{11}(n) \bmod 691$ (which is due to Ramanujan himself).

I've looked at other weights and levels with no insight. I've also looked nontraditional continued fraction expansions with alternating $\pm$ signs, and numerators other than 1, to see if I could make 691 (v. 692) appear instead.

---

The 'eigen-angles' $\theta_p$ clearly carry deep arithmetic information coming from Galois representations, as others on this site can explain much better than I. But can something be proven about these continued fraction expansions? Anything that's true is likely to be quite deep, so a more realistic (but vague) question is:

Are there other examples of this kind of 'numerology'?

Answer 1

Since the OP asked for other examples of this kind of numerology,I will give another one to support his observation

The function $\cos(\theta_{11})$ has the following closed form

$\cos(\theta_{11})=\frac{\sigma_{1}(11)}{22\sqrt{11}}-\frac{12\sum_{k=1}^{10} (2178k^2-572k^3+35k^4)\sigma_{1}(11-k)\sigma_{1}(k)}{161051\sqrt{11}}\tag1$

and continued fraction

$\cos(\theta_{11})=\{0;1,1,572,3,2,1,2,1,2,2,4,3,1,6,\dots\}\tag2$

where we clearly see $572$ appearing both as a coefficient in the sum $(1)$ and largest partial quotient in the first few partial quotients of the continued fraction $(2)$

Is this a coincidence?

Edited:03 Sep 2017

And also

$2\cos(\theta_{11})=\frac{\sigma_{1}(11)}{11\sqrt{11}}-\frac{12\sum_{k=1}^{10} (4356k^2-1144k^3+70k^4)\sigma_{1}(11-k)\sigma_{1}(k)}{161051\sqrt{11}}\tag3$

with the following continued fraction

$2\cos(\theta_{11})=\{1;1145,1,1,2,6,2,2,1,1,1,1,1,2,3,\dots\}\tag4$

where $1145$ is the 0th partial qotient in the continued fraction $(4)$ and $1144$ appears as a coefficient in the formula $(3)$

Remark: Identity $(1)$ is a special case of the identity found in A000594 OEIS when $n=11$

$\tau(n)=n^4\sigma_{1}(n)-24\sum_{k=1}^{n-1} (18n^2k^2-52nk^3+35k^4)\sigma_{1}(n-k)\sigma_{1}(k)$

a formula which was first proved by D. Niebur in A formula for Ramanujan's $\tau$-function, Illinois J. Math, $19$ (1975), 448-449.