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(Cross-posted from stackexchange: https://math.stackexchange.com/questions/1976296/what-is-known-about-the-average-of-the-partial-quotients-in-the-fundamental-peri)

I'd like references concerning the growth rate of the average of the entries in the fundamental period of the simple continued fraction expansion of $\sqrt{n}$. I'm especially interested in a lower bound, and in particular I wonder if something like the following conjecture is known:

Conjecture: For each nonsquare positive integer $n$, let $a_n$ be the average of the entries of the fundamental period of $\sqrt{n}$. Then

$$ a_n = \Omega(\ln n) $$

(using the Knuth definition of $\Omega$)

I have conducted a search through all $n$ between $2$ and $1.5 \times 10^8$. The successive minima of $a_n/ln(n)$ over this range are as follows (with pairs $(n,a_n/ln(n))$

(2,2.885)

(3, 1.365)

(7, 0.899)

(13, 0.780)

(21, 0.766)

(44, 0.661)

(115, 0.653)

(190, 0.626)

(244, 0.602)

(397609, 0.600)

(811924, 0.598)

(940801, 0.595)

(4861081, 0.594)

(6868801, 0.593)

(11468521, 0.591)

(13981081, 0.590)

(70023409, 0.589)

Edit: the originally conjecture I gave was a classic example of overfitting. Nevertheless, since I am interested in a lower bound, the data above made me concerned that in fact $a_n = o(\ln n)$, hence my reason for complicating the expression in the conjecture. But a little bit of computation supports well the following weakening of the above conjecture: $a_n = \omega( \ln n / \ln (\ln n)^{\varepsilon})$ for every $\varepsilon > 0$.

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    $\begingroup$ I should say that using numerical data to support even the appearance of $\log\log n$ in a formula is pretty much impossible: the function grows so slowly that $\log\log$ of the number of particles in the universe is only about $5.2$. In that vein, imagining that numerical data supports a specific expression with $(\log \log n)^{1/2}$, or (for heaven's sake) $(\log\log\log n)^{1/3}$, is pretty optimistic. Much more helpful evidence would be some indication of what heuristic led to the expression to begin with. $\endgroup$ – Greg Martin Oct 22 '16 at 19:08
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Partial answer.

The average can be as as large as $\sqrt{2n}$ infinitely often.

From mathworld (6)

$$ \sqrt{a^2+1}=[a;\overline{2a}]$$

There are similar identities with short period.

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Also a partial answer.

Partial quotients in periods of quadratic irrationals satisfy the Gauss-Kuz’min law (in average over quadratic irrationals with the length $\le N$), see Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals. So you can try to analise you conjecture assuming this property for individual $n$ with long period.

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