# Lower bounds for split primes in Real quadratic fields

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?

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!

This is a bit of a follow up to Franz's answer, but is also a bit different.

Dirichlet's class number formula tells us that $$hR = \frac{\mu(K)\sqrt{|D|}}{2^r(2\pi)^s}L(1,\chi)$$

A particular case of a famous result of Siegel says that for some positive constant $c$ $$L(1, \chi) > c|D|^{-1/4}$$

Combine to get that for some constant $c'$ $$hR > c'\sqrt{|D|}$$

Notice that if $|D| > 2^{48}$ then $lg(|D|/4) < \sqrt{|D|}$, so if $|D| > max\{2^{48},c'^{-8}\}$ then $$2^{hR} > 2^{c' \sqrt{|D|}} > 2^{c'c'^{-1}\sqrt{D}} > |D|/4$$ as required.

• The $c$ of Siegel is ineffective. Instead via $L$-functions you get $2^{hR}= 2^{2L(1\,chi)\sqrt d}$, so you need $(2\log 2)L(1,\chi)\sqrt D\ge\log (D/4)$ if I am correct. But this is not known effectively, as the best bounds are from the Gross-Zagier theorem and Goldfeld's work, and would be $\log(D)^{1-\epsilon}$ on the right. May 7, 2010 at 4:16
• True. But at least we know one exists. May 7, 2010 at 6:02