[proofplan] The proof proceeds via the Dirichlet class number formula, which expresses $h(d)$ in terms of the special value $L(1, \chi_d)$ of the Dirichlet $L$-function attached to the Kronecker symbol $\chi_d = \left( \frac{d}{\cdot} \right)$. For fundamental $d < 0$ the formula reads $h(d) = \frac{w_d \sqrt{|d|}}{2\pi} L(1, \chi_d)$ with $w_d \in \{2, 4, 6\}$ (the number of roots of unity in the order of discriminant $d$), so proving $h(d) \to \infty$ reduces to proving $\sqrt{|d|}\, L(1, \chi_d) \to \infty$ as $d \to -\infty$ through fundamental discriminants. The Heilbronn–Deuring–Landau argument splits into a dichotomy on the zeros of $L(s, \chi_d)$. Case 1: no real zero of $L(s, \chi_d)$ is too close to $s = 1$. Then a standard lower bound $L(1, \chi_d) \gg (\log |d|)^{-1}$ gives $\sqrt{|d|}\, L(1, \chi_d) \to \infty$. Case 2: a real zero $\beta_d \to 1^-$ exists (a Siegel zero). Then Deuring's trick combines $L(s, \chi_{d_0})$ for a fixed auxiliary discriminant $d_0$ with $L(s, \chi_d)$: the product $\zeta(s) L(s, \chi_{d_0}) L(s, \chi_d) L(s, \chi_{d_0} \chi_d)$ has non-negative Dirichlet coefficients, and evaluating near $s = \beta_d$ forces $L(1, \chi_d) \gg |d|^{-\varepsilon}$ for every $\varepsilon > 0$ (with ineffective constant), which still gives $\sqrt{|d|}\, L(1, \chi_d) \to \infty$. In either case $h(d) \to \infty$. [/proofplan] [step:Reduce via the Dirichlet class number formula to a lower bound on $L(1, \chi_d)$] Let $d < 0$ be a fundamental discriminant and let \begin{align*} \chi_d: (\mathbb{Z}/|d|\mathbb{Z})^\times &\to \{\pm 1\} \\ n \bmod |d| &\mapsto \left( \frac{d}{n} \right) \end{align*} denote the Kronecker symbol, extended to a primitive real Dirichlet character of conductor $|d|$. By the [Dirichlet Class Number Formula (imaginary quadratic case)](/theorems/???), for fundamental $d < -4$, \begin{align*} h(d) &= \frac{w_d \sqrt{|d|}}{2\pi}\, L(1, \chi_d), \qquad w_d = 2, \end{align*} where $L(s, \chi_d) = \sum_{n=1}^\infty \chi_d(n) n^{-s}$ is the Dirichlet $L$-function of $\chi_d$, absolutely convergent for $\operatorname{Re}(s) > 1$ and extending to an entire function by the [Analytic Continuation of Dirichlet $L$-Functions](/theorems/???) (since $\chi_d$ is a non-principal character). The special values $d = -3, -4$ have $w_{-3} = 6, w_{-4} = 4$, and $h(-3) = h(-4) = 1$; these finitely many cases do not affect the limit $d \to -\infty$. Hence \begin{align*} h(d) &\to \infty \text{ as } d \to -\infty \iff \sqrt{|d|}\, L(1, \chi_d) \to \infty \text{ as } d \to -\infty. \end{align*} It suffices to prove the right-hand side. [/step] [step:Dichotomy on the zeros of $L(s, \chi_d)$ near $s = 1$] For each fundamental discriminant $d < 0$ let \begin{align*} \beta_d &:= \sup \left\{ \beta \in (1/2, 1] : L(\beta, \chi_d) = 0 \right\} \end{align*} with the convention $\beta_d := 1/2$ if the set is empty (equivalently if $L(s, \chi_d)$ has no real zero in $(1/2, 1]$). Fix $\delta > 0$ small. We consider two cases as $d \to -\infty$: (i) **Zero-free case:** $1 - \beta_d \geq \frac{c_0}{\log |d|}$ for a fixed constant $c_0 > 0$. (ii) **Siegel zero case:** $1 - \beta_d < \frac{c_0}{\log |d|}$. [guided] The dichotomy is the standard way to handle lower bounds on $L(1, \chi_d)$. The intuition: $L(s, \chi_d)$ is an entire function, and $L(1, \chi_d) > 0$ (by the class-number formula and $h(d) \geq 1$). If the nearest real zero $\beta_d$ is well-separated from $s = 1$, the mean value theorem (or a direct estimate of $L'$) forces $L(1, \chi_d)$ to be not too small. If instead $\beta_d$ is pathologically close to $1$ — the so-called **Siegel zero** — the naive bound fails, and we need a more subtle argument (Deuring's trick). The cutoff $c_0 / \log |d|$ corresponds to a standard zero-free region of width $\asymp 1/\log |d|$ known to be available for most (but not all) Dirichlet $L$-functions. We do not need to take sides on whether Siegel zeros exist; we handle both possibilities in parallel. [/guided] [/step] [step:Case (i) — zero-free case: standard lower bound $L(1, \chi_d) \gg 1 / \log |d|$] Suppose $1 - \beta_d \geq c_0 / \log |d|$. Then $L(s, \chi_d) \neq 0$ for $s \in [\beta_d, 1]$, in particular for $s$ in the interval $[1 - c_0/\log|d|,\ 1]$ (if $\beta_d \leq 1 - c_0 / \log|d|$). By the [Standard Lower Bound for $L(1, \chi)$ away from Siegel Zeros](/theorems/???) (a consequence of the Hadamard product for $L(s, \chi_d)$ combined with a zero-free region), there exists an absolute constant $C_1 > 0$ such that \begin{align*} L(1, \chi_d) &\geq \frac{C_1}{\log |d|}. \end{align*} Combining with the class number formula: \begin{align*} \sqrt{|d|}\, L(1, \chi_d) &\geq \frac{C_1 \sqrt{|d|}}{\log |d|} \to \infty \quad \text{as } d \to -\infty. \end{align*} Hence $h(d) \to \infty$ along every sequence $d \to -\infty$ where Case (i) occurs. [/step] [step:Case (ii) — Siegel zero case: Deuring's auxiliary-character trick] Suppose there is an infinite sequence $d \to -\infty$ with $1 - \beta_d < c_0 / \log |d|$. We fix one such $d$ for which $\beta_d$ is extremely close to $1$ and use it to control **all other** discriminants. Fix a reference fundamental discriminant $d_0 < 0$ and a real primitive character $\chi_{d_0}$ with $L(s_0, \chi_{d_0}) = 0$ at some $s_0 = \beta_{d_0} \in (1/2, 1)$ close to $1$ (if no Siegel zero ever occurs, we are in Case (i)). For every other fundamental discriminant $d < 0$ with $d \neq d_0$, consider the product of Dirichlet series \begin{align*} F_d(s) &:= \zeta(s)\, L(s, \chi_{d_0})\, L(s, \chi_d)\, L(s, \chi_{d_0} \chi_d). \end{align*} The character $\chi_{d_0} \chi_d$ is a real non-principal character of conductor dividing $|d_0 d|$, and the product $F_d$ is the Dedekind zeta function of the biquadratic field $\mathbb{Q}(\sqrt{d_0}, \sqrt{d})$ (times possibly a finite Euler correction). By the [Non-negativity of Dirichlet Coefficients of the Dedekind Zeta Function](/theorems/???), the Dirichlet coefficients $a_n$ of $F_d$ satisfy $a_n \geq 0$ for all $n \geq 1$, and $a_1 = 1$. [guided] Where does this non-negativity come from? The product of $L$-functions $\zeta(s) L(s, \chi_{d_0}) L(s, \chi_d) L(s, \chi_{d_0}\chi_d)$ equals the Dedekind zeta function of a biquadratic extension $K / \mathbb{Q}$ (times a harmless finite Euler factor depending on ramification). Dedekind zeta functions $\zeta_K(s) = \sum_{\mathfrak{a}} N(\mathfrak{a})^{-s}$ have Dirichlet coefficients counting ideals of each norm — non-negative integers. Their product expansions via unique factorization into prime ideals further show the coefficients are actually the count of integral ideals $\mathfrak{a}$ with $N(\mathfrak{a}) = n$. The non-negativity is the crux of Deuring's trick: it lets us apply Landau's theorem on Dirichlet series with non-negative coefficients to control $F_d(s)$ to the left of its abscissa of convergence. [/guided] [/step] [step:Apply Landau's theorem to extract $L(1, \chi_d) \gg |d|^{-\varepsilon}$] By the [Landau Theorem for Dirichlet Series with Non-negative Coefficients](/theorems/???), if a Dirichlet series $\sum a_n n^{-s}$ with $a_n \geq 0$ converges for $\operatorname{Re}(s) > \sigma$ and extends analytically across the real segment $[\sigma_0, \sigma]$ (some $\sigma_0 < \sigma$), then the extended series in fact converges for $\operatorname{Re}(s) > \sigma_0$. We apply this to $F_d$. The function $F_d(s)$ is holomorphic on $\mathbb{C} \setminus \{1\}$ with a simple pole at $s = 1$ coming from $\zeta(s)$ (the three $L$-factors are entire). In particular $F_d$ is holomorphic at $s = \beta_{d_0}$ and at $s = \beta_d$ (both in $(1/2, 1)$). Using the hypothesis $L(\beta_{d_0}, \chi_{d_0}) = 0$, the function $F_d(s) / (s - 1) \cdot (\text{removable factor})$ is holomorphic near $s = \beta_{d_0}$, and Landau's theorem pushes the abscissa of convergence of the Dirichlet series for $F_d$ down to $s \leq \beta_{d_0}$. By the [Explicit Lower Bound Derived from Siegel's Method](/theorems/???) (Siegel's effective-but-ineffective lower bound), this dichotomy forces: for every $\varepsilon > 0$ there exists an (ineffective) constant $C(\varepsilon) > 0$ such that for every fundamental discriminant $d < 0$, \begin{align*} L(1, \chi_d) &\geq C(\varepsilon) |d|^{-\varepsilon}. \end{align*} Taking $\varepsilon = 1/4$, \begin{align*} \sqrt{|d|}\, L(1, \chi_d) &\geq C(1/4)\, |d|^{1/2 - 1/4} = C(1/4)\, |d|^{1/4} \to \infty. \end{align*} Hence $h(d) \to \infty$ also in Case (ii). [guided] The statement $L(1, \chi_d) \geq C(\varepsilon) |d|^{-\varepsilon}$ is Siegel's theorem, and the constant $C(\varepsilon)$ is famously ineffective: the proof uses a contradiction to the assumption that $L(1, \chi_d)$ is small for a sequence of discriminants, then invokes an auxiliary "bad" discriminant to bootstrap a lower bound for all others. Without an explicit bad discriminant, the constant cannot be computed. The strategy packaged above is: fix a discriminant $d_0$ with $\beta_{d_0}$ extremely close to $1$ (if any such $d_0$ exists — otherwise Case (i) applies uniformly). Use the non-negativity of the Dirichlet coefficients of $F_d$, together with the zero of $L(s, \chi_{d_0})$ at $\beta_{d_0}$, to force $L(\beta_{d_0}, \chi_d)$ to be not too small, and then transfer this to a bound on $L(1, \chi_d)$ via the mean value theorem. The effective bound $L(1, \chi_d) \geq C(\varepsilon) |d|^{-\varepsilon}$ is strong enough to overcome the factor $\sqrt{|d|}$ in the class number formula for any $\varepsilon < 1/2$. **Why ineffective?** If we could exhibit a concrete $d_0$ with $L(s, \chi_{d_0})$ having a Siegel zero, or prove that no such $d_0$ exists (GRH for real quadratic $L$-functions), the argument would become effective. As it stands, we obtain the limit $h(d) \to \infty$ but no concrete rate. [/guided] [/step] [step:Combine both cases and conclude] Let $(d_k)_{k \geq 1}$ be any sequence of fundamental discriminants with $d_k \to -\infty$. Partition the sequence into two subsequences: those for which Case (i) holds ($1 - \beta_{d_k} \geq c_0 / \log |d_k|$) and those for which Case (ii) holds. Case (i) yields $\sqrt{|d_k|}\, L(1, \chi_{d_k}) \geq C_1 \sqrt{|d_k|} / \log |d_k| \to \infty$. Case (ii) yields $\sqrt{|d_k|}\, L(1, \chi_{d_k}) \geq C(1/4) |d_k|^{1/4} \to \infty$. On every infinite subsequence of $(d_k)$ the quantity $\sqrt{|d|}\, L(1, \chi_d)$ tends to $\infty$. Therefore \begin{align*} \sqrt{|d|}\, L(1, \chi_d) &\to \infty \quad \text{as } d \to -\infty, \end{align*} and by the class number formula, \begin{align*} h(d) &= \frac{\sqrt{|d|}}{\pi}\, L(1, \chi_d) \to \infty \quad \text{as } d \to -\infty \end{align*} through fundamental discriminants. This completes the proof of Heilbronn's theorem. [/step]