# Cubic, abelian analogue of a result of Mertens-Siegel

For a number field $$K/\mathbb{Q}$$, put $$h_K, R_K$$ respectively for the class number (of the ring of integers $$\mathcal{O}_K$$ of $$K$$) and the regulator of $$K$$ respectively. Moreover, put $$\zeta_K(s)$$ for the Dedekind zeta function of $$K$$. We then have the class number formula:

$$\displaystyle \lim_{s \rightarrow 1} (s-1) \zeta_K(s) = \frac{2^{r_1(K)} (2\pi)^{r_2(K)} h_K R_K}{w_K \sqrt{|\Delta_K|}},$$

where $$w_K$$ is the number of roots of unity inside $$K$$, $$r_1(K)$$ the number of real embeddings of $$K$$, $$r_2(k)$$ the number of pairs of complex embeddings, and $$\Delta_K$$ the discriminant of $$K$$.

By the Brauer-Siegel theorem, we know that if $$\mathcal{K}_n$$ is the sequence of cyclic degree $$n$$ fields over $$\mathbb{Q}$$, ordered by discriminant, then

$$\displaystyle \lim_{K \in \mathcal{K}_n} \frac{\log(h_K R_K)}{(\log |\Delta_K|)/2} = 1.$$

However, the logarithm hides a lot of the information about the precise asymptotic behaviour of $$\Delta_K$$ and $$h_K R_K$$.

For $$n = 2$$ Mertens and Siegel obtained a more precise asymptotic relation, namely the asymptotic formulae:

$$\displaystyle \sum_{-X < D < 0} h_{\mathbb{Q}(\sqrt{D})} R_{\mathbb{Q}(\sqrt{D})} = c_1 X^{3/2} + O \left(X \log X\right),$$

$$\displaystyle \sum_{0 < D < X} h_{\mathbb{Q}(\sqrt{D})} R_{\mathbb{Q}(\sqrt{D})} = c_2 X^{3/2} + O \left(X \log X\right)$$

for absolute constants $$c_1, c_2 > 0$$.

Is there an analogous formula for $$n = 3$$? That is, let $$\mathcal{C}_3$$ be the set of cyclic cubic fields. Is there an asymptotic formula for

$$\displaystyle \sum_{m \leq X} \sum_{\substack{K \in \mathcal{C}_3 \\ D_K = m^2 }} h_K R_K?$$

• How did they simplify $\sum_{0< D < X} h_{\mathbb{Q}(\sqrt{D})} R_{\mathbb{Q}(\sqrt{D})}$ ? – reuns May 7 at 18:33
• Note that these formulas were also known to Gauss: mathoverflow.net/questions/109330/… though he did not provide proofs (and IIRC he had an incorrect error term for the real quadratic case). – Alison Miller May 7 at 20:10