西格尔零点

西格尔零点、西格尔零（）、兰道-西格尔零点（）、異常零点（），是以德国数学家愛德蒙·蘭道和卡爾·西格爾命名的一種对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。

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 , which can only occur when the L-function is associated to a real Dirichlet character.

Real primitive Dirichlet characters 

For an integer , a Dirichlet character modulo  is an   satisfying the following properties:
 ()  for every , ;
 (Periodic)  for every ;
 (Support)  if .
That is,  is the lifting of a  .

The trivial character is the character modulo 1, and the principal character modulo , 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   (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  (i.e., the discriminant of a ). One way to define  is as the completely multiplicative arithmetic function determined by (for  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  of the 狄利克雷级数  defined for , where s is a . For  non-principal, this continuation is ; otherwise it has a  of   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  is equivalent (in a certain sense) to  (). Moreover, via the , we can reflect these regions through  to conclude that, with the exception of negative integers of same parity as , 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  (GRH) claims that for every , all the non-trivial zeros of  lie on the line .

定義「西格爾零點」 

The definition of Siegel zeros as presented ties it to the constant  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  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  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  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  of the  . This technique is still largely applied in modern works.

This 'repelling effect' (see ), 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 .

Siegel–Tatuzawa 定理 
In 1951,  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,  (1973) showed that Euler's list of 65  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  (see ). The precise relation between the distribution of zeros near  and arithmetic comes from :

where:
  is the  of ;
  is the number of  in  ();
  is the  of  ().
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  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 , for example, that the   of an   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 s, the same is true:

 Theorem (, 1974). Let  be a number field of degree . 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  of the extension .

"No Siegel zeros" for D < 0 
When dealing with quadratic fields, the case  tends to be elusive due to the behaviour of the fundamental unit. Thus, it is common to treat the cases  and  separately. Much more is known for the negative discriminant case:

Lower bounds for h(D) 
In 1918,  showed that "no Siegel zeros" for  implies that  (see  for comparison). This can be extended to an equivalence, as it is a consequence of Theorem 3 in – (2000):

where the summation runs over the    of discriminant . Using this, Granville and Stark showed that a certain uniform formulation of the  for number fields implies "no Siegel zeros" for negative discriminants.

In 1976,  proved the following unconditional, effective lower bound for :

Complex multiplication 
Another equivalence for "no Siegel zeros" for  can be given in terms of  for  of :

where:
  is the absolute  for number fields;
  is the ;
 .
The number  generates the  of , 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  (1993, 1998), who showed that, for an elliptic curve  with  by , we have

where  denotes the . Using the identities  and , Colmez' theorem also provides a proof for the equivalence above.

Consequences of Siegel zeros existing 
Although the  is expected to be true, since the "no Siegel zeros" conjecture remains open, it is interesting to study what consequences such severe counterexamples to the GRH would imply. Another reason to study this possibility is that the proof of certain unconditional theorems require the division into two cases: first a proof assuming no Siegel zeros exist, then another assuming Siegel zeros do exist. A classical theorem of this type is 林尼克定理 on the smallest .

The following are some examples of facts that follow from the existence of Siegel zeros.

Infinitude of twin primes 

A striking result in this direction is 's 1983 result which, following , can be stated as follows:
 Theorem (Heath-Brown, 1983). At least one of the following is true: (1) There are no Siegel zeros. (2) There are infinitely many .

Parity problem 

The parity problem in  roughly refers to the fact that sieving arguments are, generally speaking, unable to tell if an integer has an even or odd number of prime divisors. This leads to many upper bounds in sieve estimates, such as the one from the  being off by a factor of 2 from the expected value. In 2020,  showed that under the assumption of the existence of Siegel zeros, the general upper bounds for the problem of sieving intervals are optimal, meaning that the extra factor of 2 coming from the parity phenomenon would thus not be an artificial limitation of the method.

另見 
 广义黎曼猜想

參考