My broad question is as in the title.
The motivation is the following: let $\Delta$ be a positive, nonsquare integer congruent to $0$ or $1$ modulo $4$. Set $\varepsilon = 1$ if $\Delta$ is odd and $\varepsilon = 0$ if $\Delta$ is even. An earlier post
Must a certain continued fraction have "small" partial quotients?
shows that the partial quotients of the fundamental period when expanding $\frac{\sqrt{\Delta}+\varepsilon}{2}$ in a continued fraction are bounded above.
Computational evidence suggests that there is competing pressure from below. For $\Delta < 100000$, the average the partial quotients in the fundamental period is $\geq 2$ except when $\Delta = 5$, $12$, $17$, or $28$, and it seems possible that these averages will eventually exceed any given bound.
So I would like to find results about averages over fundamental periods in the literature. I am in particular looking for results about the quadratic irrationalities above, especially for a proof that they are above 2 outside of the exceptions I noted. But I am also happy to see asymptotic or ineffective bounds or results for other sorts of quadratic irrationalities.
Well, in the meantime, developed a conjecture in the case where $\Delta$ is even. In that case, $\frac{\sqrt{\Delta}+\varepsilon}{2}$ is just $\sqrt{n}$ for $n = \Delta/4$.
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 $8 \times 10^7$. 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 on $a_n$, 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 shows that data supports well the following weakening of the above conjecture: $a_n = \omega( \ln n / \ln (\ln n)^{\varepsilon})$ for every $\varepsilon > 0$.