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Majorising $h_K\mathcal{R}_K$

Majorising $h_K$

Using the influence of small primes

The $h^-_K$ of CM-fields

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Explicit bounds for class numbers

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Let $K$ be a number field of degree $n\ge2$, signature $(r_1,r_2)$, absolute value of discriminant $d_K$, class number $h_K$, regulator $\mathcal{R}_K$ and $w_K$ the number of roots of unity in $K$. We further denote by $\kappa_K$ the residue at $s=1$ of the Dedekind zeta-function $\zeta_K(s)$ attached to $K$.

Estimating $h_K$ is a long-standing problem in algebraic number theory. One of the classic way is the use of the so-called analytic class number formula stating that $$ h_K\mathcal{R}_K=\frac{w_K \sqrt{d_K}}{2^{r_1}(2\pi)^{r_2}}\kappa_K $$ and to use Hecke's integral representation of the Dedekind zeta function to bound $\kappa_K$. This is done in and in with additional properties of log-convexity of some functions related to $\zeta_K$ and enabled Louboutin to reach the following bound: $$ h_K\mathcal{R}_K \le\frac{w_K}{2}\left(\frac{2}{\pi}\right)^{r_2} \left(\frac{e\log d_K}{4n-4}\right)^{n-1}\sqrt{d_K}. $$ Furthermore, if $\zeta_K(\beta)=0$ for some $\tfrac12\le \beta< 1$, then we have $$ h_K\mathcal{R}_K \le(1-\beta)w_K\left(\frac{2}{\pi}\right)^{r_2} \left(\frac{e\log d_K}{4n}\right)^{n}\sqrt{d_K}. $$ When $K$ is abelian, then the residue $\kappa_K$ may be expressed as a product of values at $s=1$ of $L$-functions associated to primitive Dirichlet characters attached to $K$. On using estimates for such $L$-functions from , we get for instance $$ h_K\mathcal{R}_K \le \frac{w_K}{2}\left(\frac{2}{\pi}\right)^{r_2} \left(\frac{\log d_K}{4n-4}+\frac{5-\log 36}{4}\right)^{n-1}\sqrt{d_K}. $$ Note that the constant $\frac14(5-\log 36)=0.354\cdots$ can be improved upon in many cases. For instance, when $K$ is abelian and totally real (i.e. $r_2=0$), a result from implies that the constant may be replaced by 0, so that $$ h_K\mathcal{R}_K \le \left(\frac{\log d_K}{4n-4}\right)^{n-1}\sqrt{d_K}. $$ One may also estimate $h_K$ alone, without any contamination by the regulator since this contamination is often difficult to control, see . In this case, one rather uses explicit bounds for the Piltz-Dirichlet divisor functions $\tau_n$ (see and ) and get $$ h_K\le \frac{M_K}{(n-1)!} \left(\frac{\log\bigl(M^2_Kd_K\bigr)}{2}+n-2\right)^{n-1} \sqrt{d_K} $$ as soon as $$ n\ge 3,\quad d_K\ge 139 M_K^{-2} \quad\text{where}\quad M_K=(4/\pi)^{r_2}n!/n^n. $$ The constant $M_K$ is known as the Minkowski constant of K. In we find the following.
Theorem (2021)

Ler $K$ be a quartic number field with class number $h_K$ and Minkowski bound $b$. Then if $b\ge 193$, we have $h_K\le (1/3) x(\log x)^3$.

It is explained in how the behavior of certain small primes may subtantially improve on the previous bounds. To make things more significant, define, for a rational prime $p$, $$ \Pi_K(p)=\prod_{\mathfrak{p}|p}\left(1-\frac{1}{\mathcal{N}_K(\mathfrak{p})}\right)^{-1}. $$ From , we have among other things $$ h_K\mathcal{R}_K \le\frac{w_K}{2} \left(\frac{2}{\pi}\right)^{r_2} \frac{\Pi_K(2)}{\Pi_{\mathbb{Q}}(2)^n} \left(\frac{e\log d_K}{4n-4}\times e^{n\log 4/\log d_K} \right)^{n-1}\sqrt{d_K} $$ where $K$ is any number field of degree $n\ge3$. In particular, when $2$ is inert in $K$, then $$ h_K\mathcal{R}_K \le\frac{w_K}{2(2^n-1)} \left(\frac{2}{\pi}\right)^{r_2} \left(\frac{e\log d_K}{4n-4}\times e^{n\log 4/\log d_K} \right)^{n-1}\sqrt{d_K}. $$ Let $K$ be here a CM-field of degree $2n > 2$, i.e. a totally complex quadratic extension $K$ of its maximal totally real subfield $K^+$. it is well known that $h_{K^+}$ divides $h_K$. The quotient is denoted by $h^-_K$ and is called the relative class number of $K$. The analytic class number formula yields $$ h^-_K=\frac{Q_Kw_K}{(2\pi)^n} \left(\frac{d_K}{d_{K^+}}\right)^{1/2} \frac{\kappa_K}{\kappa_{K^+}} = \frac{Q_Kw_K}{(2\pi)^n} \left(\frac{d_K}{d_{K^+}}\right)^{1/2} L(1,\chi) $$ where $\chi$ is the quadratic character of degree 1 attached to the extension $K/K^+$ and $Q_K\in\{1,2\}$ is the Hasse unit index of $K$. Here are three results originating in this formula. From :
Theorem (2000)

We have $$ h^-_K \le 2Q_Kw_K\left(\frac{d_K}{d_{K^+}}\right)^{1/2} \left( \frac{e\log(d_K/d_{K^+})}{4\pi n} \right)^n. $$

From :
Theorem (2003)

Assume that $(\zeta_K/\zeta_{K^+})(\sigma)\ge0$ whenever $0 < \sigma < 1$. Then we have $$ h^-_K \ge \frac{Q_Kw_K}{\pi e \log d_K} \left(\frac{d_K}{d_{K^+}}\right)^{1/2} \left( \frac{n-1}{\pi e\log d_K} \right)^{n-1}. $$

Again from :
Theorem (2003)

Let $c=6-4\sqrt{2}=0.3431\cdots$. Assume that $d_K\ge 2800^n$ and that either $K$ does not contain any imaginary quadratic subfield, or that the real zeros in the range $1-\frac{c}{\log d_N}\le \sigma < 1$ of the Dedekind zeta-functions of the imaginary quadratic subfields of $K$ are nor zeros of $\zeta_K(s)$, where $N$ is the normal closure of $K$. Then we have $$ h^-_K \ge \frac{cQ_Kw_K}{4ne^{c/2}[N:\mathbb{Q}]} \left(\frac{d_K}{d_{K^+}}\right)^{1/2} \left( \frac{n}{\pi e\log d_K} \right)^{n}. $$

And a third result from :
Theorem (2003)

Assume $n > 2$, $d_K > 2800^n$ and that $K$ contains an imaginary quadratic subfield $F$ such that $\zeta_F(\beta)=\zeta_K(\beta)=0$ for some $\beta$ satisfying $1-\frac{2}{\log d_K}\le \beta < 1$. Then we have $$ h^-_K \ge \frac{6}{(\pi e)^2} \left(\frac{d_K}{d_{K^+}}\right)^{1/2-1/n} \left( \frac{n}{\pi e\log d_K} \right)^{n-1}. $$

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