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?