西格爾零點

西格爾零點、西格爾零（Siegel zero）、蘭道-西格爾零點（Landau-Siegel zero）、異常零點（exceptional zero），是以德國數學家愛德蒙·蘭道和卡爾·西格爾命名的一種對廣義黎曼假設潛在反例的解析數論猜想，是關於與二次域相關的狄利克雷L函數的零點。粗略說，這些可能的零點在可量化的意義上可以非常接近s = 1。

## 動機和定義

狄利克雷L函數有與黎曼ζ函數相似的無零點區域。

The way in which Siegel zeros appear in the theory of Dirichlet L-functions is as potential exceptions to the classical zero-free regions, which can only occur when the L-function is associated to a real Dirichlet character.

### Real primitive Dirichlet characters

For an integer q ≥ 1, a Dirichlet character modulo q is an arithmetic function $\chi:\mathbb{Z}\to\mathbb{C}$ satisfying the following properties:

- (Completely multiplicative) $\chi(mn) = \chi(m)\chi(n)$ for every m, n;
- (Periodic) $\chi(n+q) = \chi(n)$ for every n;
- (Support) $\chi(n) = 0$ if $\mathrm{gcd}(n,q) > 1$.

That is, χ is the lifting of a homomorphism $\widetilde{\chi}:(\mathbb{Z}/q\mathbb{Z})^{\times} \to \mathbb{C}^{*}$.

The trivial character is the character modulo 1, and the principal character modulo q, denoted $\chi_0~(\mathrm{mod}~q)$, is the lifting of the trivial homomorphism $(\mathbb{Z}/q\mathbb{Z})^{\times}\ni a \mapsto 1 \in \mathbb{C}^{*}$.

A character $\chi~(\mathrm{mod}~{q})$ is called imprimitive if there exists some integer $d\neq q$ with $d\mid q$ such that the induced homomorphism $\widetilde{\chi}:(\mathbb{Z}/q\mathbb{Z})^{\times} \to \mathbb{C}^{*}$ factors as

$$(\mathbb{Z}/q\mathbb{Z})^{\times}\twoheadrightarrow (\mathbb{Z}/d\mathbb{Z})^{\times} \xrightarrow{\widetilde{\chi'}} \mathbb{C}^{*}$$

for some character $\chi'~(\mathrm{mod}~{d})$; otherwise, $\chi~(\mathrm{mod}~{q})$ is called primitive.

A character $\chi$ is real (or quadratic) if it equals its complex conjugate $\overline{\chi}$ (defined as $\overline{\chi}(n) := \overline{\chi(n)}$), or equivalently if $\chi^2 = \chi_0$. The real primitive Dirichlet characters are in one-to-one correspondence with the 克羅內克符號s $(D|\,\cdot\,): \mathbb{Z} \to \{-1,0,1\}$ for $D\in\mathbb{Z}$ a fundamental discriminant (i.e., the discriminant of a quadratic number field). One way to define $(D|\,\cdot\,)$ is as the completely multiplicative arithmetic function determined by (for p prime):

$$\bigg(\frac{D}{p}\bigg)=\begin{cases}1, &(p)\text{ splits in } \mathbb{Q}(\sqrt{D}), \\ -1, &(p)\text{ is inert } \cdots, \\ 0, &(p)\text{ ramifies } \cdots, \end{cases} \quad \bigg(\frac{D}{-1}\bigg) = \text{sign of } D.$$

It is thus common to write $\chi_D := (D|\,\cdot\,)$, which are real primitive characters modulo $|D|$.

### Classical zero-free regions

The Dirichlet L-function associated to a character $\chi~(\mathrm{mod}~q)$ is defined as the analytic continuation of the 狄利克雷級數 $L(s,\chi) = \sum_{n\geq 1} \chi(n) n^{-s}$ defined for $\mathrm{Re}(s)>1$, where s is a complex variable. For $\chi$ non-principal, this continuation is entire; otherwise it has a simple pole of residue $\prod_{p\mid q}(1-p^{-1})$ at s = 1 as its only singularity. For $\mathrm{Re}(s)>1$, Dirichlet L-functions can be expanded into an 歐拉乘積 $L(s,\chi) = \prod_{p} (1 - \chi(p)p^{-s})^{-1}$, from where it follows that $L(s,\chi)$ has no zeros in this region. The prime number theorem for arithmetic progressions is equivalent (in a certain sense) to $L(1+it,\chi) \neq 0$ ($\forall t\in\mathbb{R}$). Moreover, via the functional equation, we can reflect these regions through $s\mapsto 1-s$ to conclude that, with the exception of negative integers of same parity as χ, all the other zeros of $L(s,\chi)$ must lie inside $\{0<\mathrm{Re}(s)<1\}$. This region is called the critical strip, and zeros in this region are called non-trivial zeros.

The classical theorem on zero-free regions (Grönwall, Landau, Titchmarsh) states that there exists a(n) (effectively computable) real number $A>0$ such that, writing $s=\sigma + it$ for the complex variable, the function $L(s,\chi)$ has no zeros in the region

$$\sigma > 1 - \frac{A}{(\log q(|t|+2))}$$

if $\chi~(\mathrm{mod}~q)$ is non-real. If $\chi$ is real, then there is at most one zero in this region, which must necessarily be real and simple. This possible zero is the so-called Siegel zero.

The Generalized Riemann Hypothesis (GRH) claims that for every $\chi~(\mathrm{mod}~q)$, all the non-trivial zeros of $L(s,\chi)$ lie on the line $\mathrm{Re}(s)=\frac{1}{2}$.

### 定義「西格爾零點」

Is there $\delta >0$ for which $L(\sigma,\chi_D)\neq 0$ for every fundamental discriminant D provided $1-\frac{\delta}{\log|D|} < \sigma < 1$?

The definition of Siegel zeros as presented ties it to the constant A in the zero-free region. This often makes it tricky to deal with these objects, since in many situations the particular value of the constant A is of little concern. Hence, it is usual to work with more definite statements, either asserting or denying, the existence of an infinite family of such zeros, such as in:

- Conjecture ("no Siegel zeros"): If $\beta_D$ denotes the largest real zero of $L(s,\chi_D)$, then $1-\beta_D \gg \frac{1}{\log|D|}.$

The possibility of existence or non-existence of Siegel zeros has a large impact in closely related subjects of number theory, with the "no Siegel zeros" conjecture serving as a weaker (although powerful, and sometimes fully sufficient) substitute for GRH (see below for an example involving Siegel–Tatuzawa's Theorem and the idoneal number problem). An equivalent formulation of "no Siegel zeros" that does not reference zeros explicitly is the statement:

$$\frac{L'}{L}(1,\chi_D) = O(\log|D|).$$

The equivalence can be deduced for example by using the zero-free regions and classical estimates for the number of non-trivial zeros of $L(s,\chi)$ up to a certain height.

## Landau–Siegel estimates

The first breakthrough in dealing with these zeros came from Landau, who showed that there exists an effectively computable constant B > 0 such that, for any $\chi_D$ and $\chi_{D'}$ real primitive characters to distinct moduli, if $\beta, \beta'$ are real zeros of $L(s,\chi_D), L(s,\chi_{D'})$ respectively, then

$$\min\{\beta,\beta'\} < 1- \frac{B}{\log|DD'|}.$$

This is saying that, if Siegel zeros exist, then they cannot be too numerous. The way this is proved is via a 'twisting' argument, which lifts the problem to the Dedekind zeta function of the biquadratic field $\mathbb{Q}(\sqrt{D},\sqrt{D'})$. This technique is still largely applied in modern works.

This 'repelling effect' (see Deuring–Heilbronn phenomenon), after more careful analysis, led Landau to his 1936 theorem, which states that for every $\varepsilon > 0$, there is $C(\varepsilon)\in\mathbb{R}_{+}$ such that, if $\beta$ is a real zero of $L(s,\chi_D)$, then $\beta < 1 - C(\varepsilon)|D|^{-\frac{3}{8} - \varepsilon}$. However, in the same year, in the same issue of the same journal, Siegel directly improved this estimate to

$$\beta < 1 - C(\varepsilon)|D|^{-\varepsilon}.$$

Both Landau's and Siegel's proofs provide no explicit way to calculate $C(\varepsilon)\in\mathbb{R}_{+}$, thus being instances of an ineffective result.

### Siegel–Tatuzawa 定理

In 1951, T. Tatuzawa proved an 'almost' effective version of Siegel's theorem, showing that for any fixed $0 < \varepsilon < \frac{1}{11.2}$, if $|D| > e^{1/\varepsilon}$ then

$$L(1,\chi_D) > 0.655|D|^{-\varepsilon},$$

with the possible exception of at most one fundamental discriminant. Using the 'almost effectivity' of this result, P. J. Weinberger (1973) showed that Euler's list of 65 idoneal numbers is complete except for at most one element.

## Relation to quadratic fields

Siegel zeros often appear as more than an artificial issue in the argument for deducing zero-free regions, since zero-free region estimates enjoy deep connections to the arithmetic of quadratic fields. For instance, the identity $\zeta_{\mathbb{Q}(\sqrt{D})}(s) = \zeta(s) L(s,\chi_D)$ can be interpreted as an analytic formulation of quadratic reciprocity (see Artin reciprocity law §Statement in terms of L-functions). The precise relation between the distribution of zeros near s = 1 and arithmetic comes from Dirichlet's class number formula:

$$L(1,\chi_D) = \begin{cases} \dfrac{2 \pi}{w_D \sqrt{|D|}} \, h(D), &\text{if } D < 0 \\[.5em] \dfrac{\log\varepsilon_D}{\sqrt{D}}\, h(D), &\text{if } D > 0, \end{cases}$$

where:

- $h(D)$ is the ideal class number of $\mathbb{Q}(\sqrt{D})$;
- $w_D$ is the number of roots of unity in $\mathbb{Q}(\sqrt{D})$ (D 0).

This way, estimates for the largest real zero of $L(s,\chi_D)$ can be translated into estimates for $L(1,\chi_D)$ (via, for example, the fact that $|L'(\sigma,\chi)| = O(\log^2 q)$ for $1-\frac{1}{\log q} \leq \sigma \leq 1$), which in turn become estimates for $h(D)$. Classical works in the subject treat these three quantities essentially interchangeably, although the case D > 0 brings additional complications related to the fundamental unit.

### Siegel zeros as 'quadratic phenomena'

There is a sense in which the difficulty associated to the phenomenon of Siegel zeros in general is entirely restricted to quadratic extensions. It is a consequence of the Kronecker–Weber theorem, for example, that the Dedekind zeta function $\zeta_{K}(s) = \sum_{I\subseteq \mathfrak{O}_K} [\mathfrak{O}_K: I]^{-s}$ of an abelian number field $K/\mathbb{Q}$ can be written as a product of Dirichlet L-functions. Thus, if $\zeta_{K}(s)$ has a Siegel zero, there must be some subfield $F\subseteq K$ with $[F:\mathbb{Q}] = 2$ such that $\zeta_{F}(s)$ has a Siegel zero.

While for the non-abelian case $\zeta_{K}(s)$ can only be factored into more complicated Artin L-functions, the same is true:

- Theorem (Stark, 1974). Let $K/\mathbb{Q}$ be a number field of degree n > 1. There is a constant $c(n)$ ($= 4$ if $K/\mathbb{Q}$ is normal, $= 4n!$ otherwise) such that, if there is a real $\beta$ in the range

$$1 - \frac{c(n)}{\log|\Delta_K|} \leq \beta < 1$$

with $\zeta_K(\beta) = 0$, then there is a quadratic subfield $F\subseteq K$ such that $\zeta_{F}(\beta)=0$. Here, $\Delta_K$ is the field discriminant of the extension $K/\mathbb{Q}$.

## "No Siegel zeros" for D < 0

When dealing with quadratic fields, the case $D>0$ tends to be elusive due to the behaviour of the fundamental unit. Thus, it is common to treat the cases $D<0$ and $D>0$ separately. Much more is known for the negative discriminant case:

### Lower bounds for h(D)

In 1918, Hecke showed that "no Siegel zeros" for $D<0$ implies that $h(D) \gg \sqrt{|D|}(\log|D|)^{-1}$ (see Class number problem for comparison). This can be extended to an equivalence, as it is a consequence of Theorem 3 in Granville–Stark (2000):

$$\text{``No Siegel zeros'' for } D<0 \quad\iff\quad h(D) \gg \frac{\sqrt{|D|}}{\log|D|}\sum_{(a,b,c)} \frac{1}{a},$$

where the summation runs over the reduced binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $D$. Using this, Granville and Stark showed that a certain uniform formulation of the abc conjecture for number fields implies "no Siegel zeros" for negative discriminants.

In 1976, D. Goldfeld proved the following unconditional, effective lower bound for $h(D)$:

$$h(D) \gg \prod_{p\mid D} \bigg(1-\frac{2\sqrt{p}}{p+1}\bigg)\, \log|D|.$$

### Complex multiplication

Another equivalence for "no Siegel zeros" for $D<0$ can be given in terms of upper bounds for heights of singular moduli:

$$h(j(\tau_D)) \ll \log|D|,$$

where:

- $h$ is the absolute logarithmic naïve height for number fields;
- $j$ is the j-invariant function;
- $\tau_D := (D+\sqrt{D})/2$.

The number $j(\tau_D)$ generates the Hilbert class field of $\mathbb{Q}(\sqrt{D})$, which is its maximal unramified abelian extension. This equivalence is a direct consequence of the results in Granville–Stark (2000), and can be seen in C. Táfula (2019).

A precise relation between heights and values of L-functions was obtained by P. Colmez (1993, 1998), who showed that, for an elliptic curve $E_D/\mathbb{C}$ with complex multiplication by $\mathbb{Z}[\tau_D]$, we have

$$-2 h_{\mathrm{Fal}}(E_D) - \frac{1}{2} \log|D| = \frac{L'}{L}(0,\chi_D) + \log 2\pi,$$

where $h_{\mathrm{Fal}}$ denotes the Faltings height.

$-2 h_{\mathrm{Fal}}(X) - \frac{1}{2}\log D$

Using the identities $h_{\mathrm{Fal}}(E_D) = \frac{1}{12}h(j(\tau_D)) + O(\log h(j(\tau_D)))$ and $\frac{L'}{L}(1,\chi_D) = -\frac{L'}{L}(0,\chi_D) - \log|D| + \log 2\pi + \gamma$, Colmez' theorem also provides a proof for the equivalence above.

## 西格爾零點存在所造成的結果

盡管一般預期廣義黎曼猜想是對的，但由於「西格爾零點不存在」的猜想依舊開放之故，因此研究「假如廣義黎曼猜想如此的反例存在的話，會有什麼結果」，也是一個令人感興趣的題目。

另一個研究如此可能性的理由，是迄今為止，部分的無條件證明要分成兩部分：第一部分是假定西格爾零點不存在，第二部分是假定西格爾零點存在，並證明說想要的定理在這兩種狀況下都成立。一個如此為之的經典案例是關於算數數列中最小的質數的林尼克定理。

以下是在西格爾零點存在的狀況下，所會造成的結果。

### 存在無限多個孿生質數

羅傑·希斯-布朗在1983年做出的一個令人驚訝的結果，用陶哲軒的話，可如下陳述：

- 定理（Heath-Brown, 1983）：以下兩個命題至少有一為真：(1)不存在西格爾零點；(2)存在有無限多的孿生質數。

換句話說，如果(1)不成立，也就是西格爾零點存在的話，那(2)就必須成立；反之若(1)成立，也就是西格爾零點不存在的話，那(2)是否成立依舊是未知數。

### 篩法的奇偶性問題

篩法的奇偶性問題指的是篩法無法顯示出篩選出的整數有奇數個或偶數個質因數這樣的問題。

這使得很多運用篩法的估計，像是使用線性篩（linear sieve）做出的估計，會以一個2的因子，與預期值產生誤差。

在2020年，安德魯·關維證明說假若西格爾零點存在，那麼篩法篩選區間的一般上界就是最佳的，換句話說，在這種狀況下，奇偶性多出來的這個2的因子，就不會是篩法的人為限制。

## 另見

- 廣義黎曼猜想
- Deuring–Heilbronn phenomenon
- Class number problem
- Brauer–Siegel theorem
- Siegel–Walfisz theorem