Snippet portion:
From Iwaniec and Kowalski's Analytic Number Theory:


If the class number $h=h(D)$ is small, then there are only few
prime ideals $\bf{p}$ of degree one with small norm. Indeed, if
$p=\bf{p \bar{p}}$ with $(\bf{p},\bf{\bar{p}})=1$, then $\bf{p}^h$
is a principal ideal generated by $\frac{1}{2}(m+n\sqrt{D})$ with
$n \ne 0$, when $p^h = \frac{1}{4}(m^2-n^2 D) \ge \frac{|D|}{4}$.


Therefore the least prime $p\_1 = p\_1(D)$ with $\chi\_D(p\_1)=1$
satisfies $p\_1 \ge {(\frac{|D|}{4})}^{1/h}$.


Hence $\chi\_D(n)$ agrees with $\mu(n)$ on all squarefree numbers
$n \le {(\frac{|D|}{4})}^{1/h}$ with
$(n,{(\frac{|D|}{4})}^{1/h})=1$. This property is not likely to
hold in long segments (because $\chi\_D$ is periodic while $\mu$ is
not), therefore this suggests that h is rather large.


Question portion:
Although the above argument would not work in a Real quadratic field ($D > 0$ so the last inequality in the first paragraph does not hold), it seems that if we replace the class number h with h times the regulator this should work.


Any ideas on how to actually show this?



Lower bounds for split primes in Real quadratic fields
The following is not a full answer, but perhaps gives you an idea of how to approach the result.


Let us consider the claim
$$ p^{hR} \ge \Big(\frac{D}4\Big) $$
for the smallest noninert prime. I first show that the inequality holds whenever $p \ge 11$.
In fact, we have $h \ge 1$ and $R \ge \frac12 \log D + O(1)$: the latter claim is clear since the fundamental unit $(t+u\sqrt{D})/2$ satisfies $u \ge 1$ and $t \approx u \sqrt{D}$. Since $\log p > 2$, the claimed inequality follows.


Thus we only have to look at small primes. If $R$ is large enough, the claim holds.
If $R$ is as small as possible, the following happens. If $u = 1$, the discriminant must have a very special form. One possibility is $D = 4n^2 +1$. Here the fundamental unit is $\varepsilon = 2n+\sqrt{D}$ (unless $n = 1$), so $R \approx \frac12 \log D$. On the other hand, "Davenport's Lemma" shows that the minimal nontrivial norm is $n = N((2n+1+\sqrt{D})/2)$; see


* Ankeny, Chowla & Hasse,  *On the class-number of the maximal real subfield of a cyclotomic field*  J. Reine Angew. Math. 217, 217-220 (1965);


a different idea for a proof of this and similar results can be found in
* Lemmermeyer & Pethö, *Simplest cubic fields*, Manuscr. Math. 88, No.1, 53-58 (1995).


Observe that we only have to check the claimed inequality for $n \le 7$.


Since the result holds for $R$ large and very small, I would guess that its proof is within reach. Good luck!


