西格尔零点

西格尔零点、西格尔零（Siegel zero）、兰道-西格尔零点（Landau-Siegel zero）、异常零点（exceptional zero），是以德国数学家爱德蒙·兰道和卡尔·西格尔命名的一种对广义黎曼假设潜在反例的解析数论猜想，是关于与二次域相关的狄利克雷L函数的零点。粗略说，这些可能的零点在可量化的意义上可以非常接近。

**动机和定义**
狄利克雷L函数有与黎曼ζ函数相似的无零点区域。
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  satisfying the following properties:- (Completely multiplicative)  for every m, n;- (Periodic)  for every n;- (Support)  if .
That is, χ is the lifting of a homomorphism .
The trivial character is the character modulo 1, and the principal character modulo q, denoted , is the lifting of the trivial homomorphism . 
A character  is called imprimitive if there exists some integer  with  such that the induced homomorphism  factors as

for some character ; otherwise,  is called primitive. 
A character  is real (or quadratic) if it equals its complex conjugate  (defined as ), or equivalently if . The real primitive Dirichlet characters are in one-to-one correspondence with the 克罗内克符号s  for  a fundamental discriminant (i.e., the discriminant of a quadratic number field). One way to define  is as the completely multiplicative arithmetic function determined by (for p prime):

It is thus common to write , which are real primitive characters modulo .

**Classical zero-free regions**
The Dirichlet L-function associated to a character  is defined as the analytic continuation of the 狄利克雷级数  defined for , where s is a complex variable. For  non-principal, this continuation is entire; otherwise it has a simple pole of residue  at  as its only singularity. For , Dirichlet L-functions can be expanded into an 欧拉乘积 , from where it follows that  has no zeros in this region. The prime number theorem for arithmetic progressions is equivalent (in a certain sense) to  (). Moreover, via the functional equation, we can reflect these regions through  to conclude that, with the exception of negative integers of same parity as χ, all the other zeros of  must lie inside . 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  such that, writing  for the complex variable, the function  has no zeros in the region

if  is non-real. If  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 , all the non-trivial zeros of  lie on the line .

**定义「西格尔零点」**
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  denotes the largest real zero of , then 
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:

The equivalence can be deduced for example by using the zero-free regions and classical estimates for the number of non-trivial zeros of  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  and  real primitive characters to distinct moduli, if  are real zeros of  respectively, then

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 . 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 , there is  such that, if  is a real zero of , then . However, in the same year, in the same issue of the same journal, Siegel directly improved this estimate to

Both Landau's and Siegel's proofs provide no explicit way to calculate , 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 , if  then

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  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  and arithmetic comes from Dirichlet's class number formula:

where:-  is the ideal class number of ;-  is the number of roots of unity in  (D  0).
This way, estimates for the largest real zero of  can be translated into estimates for  (via, for example, the fact that  for ), which in turn become estimates for . 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  of an abelian number field  can be written as a product of Dirichlet L-functions. Thus, if  has a Siegel zero, there must be some subfield  with  such that  has a Siegel zero.
While for the non-abelian case  can only be factored into more complicated Artin L-functions, the same is true:- Theorem (Stark, 1974). Let  be a number field of degree n > 1. There is a constant  ( if  is normal,  otherwise) such that, if there is a real  in the range

with , then there is a quadratic subfield  such that . Here,  is the field discriminant of the extension .

**"No Siegel zeros" for D - dead-url=no- editor4-first=J.
*dead-url=no- language=de