西格尔零点

西格尔零点、西格尔零、兰道-西格尔零点、异常零点，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 , 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 formula_1 satisfying the following properties:
That is, is the lifting of a formula_6.
The trivial character is the character modulo 1, and the principal character modulo , denoted formula_7, is the lifting of the trivial homomorphism formula_8. 
A character formula_9 is called imprimitive if there exists some integer formula_10 with formula_11 such that the induced homomorphism formula_6 factors as
for some character formula_14; otherwise, formula_9 is called primitive. 
A character formula_16 is real (or quadratic) if it equals its formula_17 (defined as formula_18), or equivalently if formula_19. The "real primitive Dirichlet characters" are in one-to-one correspondence with the 克罗内克符号s formula_20 for formula_21 a (i.e., the discriminant of a ). One way to define formula_22 is as the completely multiplicative arithmetic function determined by (for prime):
It is thus common to write formula_24, which are real primitive characters modulo formula_25.
Classical zero-free regions.
The Dirichlet L-function associated to a character formula_26 is defined as the of the 狄利克雷级数 formula_27 defined for formula_28, where "s" is a . For formula_29 non-principal, this continuation is ; otherwise it has a of formula_30 at as its only singularity. For formula_28, Dirichlet L-functions can be expanded into an 欧拉乘积 formula_32, from where it follows that formula_33 has no zeros in this region. The is equivalent (in a certain sense) to formula_34 (formula_35). Moreover, via the , we can reflect these regions through formula_36 to conclude that, with the exception of negative integers of same parity as , all the other zeros of formula_33 must lie inside formula_38. 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 formula_39 such that, writing formula_40 for the complex variable, the function formula_33 has no zeros in the region
if formula_26 is non-real. If formula_16 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 formula_26, all the non-trivial zeros of formula_33 lie on the line formula_47.
定义「西格尔零点」.
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:
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 formula_33 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 formula_53 and formula_54 real primitive characters to distinct moduli, if formula_55 are real zeros of formula_56 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 formula_58. 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 formula_59, there is formula_60 such that, if formula_61 is a real zero of formula_49, then formula_63. 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 formula_60, 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 formula_66, if formula_67 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 formula_69 can be interpreted as an analytic formulation of (see ). The precise relation between the distribution of zeros near and arithmetic comes from :
where:
This way, estimates for the largest real zero of formula_49 can be translated into estimates for formula_78 (via, for example, the fact that formula_79 for formula_80), which in turn become estimates for formula_71. 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 formula_82 of an formula_83 can be written as a product of Dirichlet L-functions. Thus, if formula_84 has a Siegel zero, there must be some subfield formula_85 with formula_86 such that formula_87 has a Siegel zero.
While for the non-abelian case formula_84 can only be factored into more complicated s, the same is true:
"No Siegel zeros" for "D" < 0.
When dealing with quadratic fields, the case formula_101 tends to be elusive due to the behaviour of the fundamental unit. Thus, it is common to treat the cases formula_102 and formula_101 separately. Much more is known for the negative discriminant case:
Lower bounds for "h"("D").
In 1918, showed that "no Siegel zeros" for formula_102 implies that formula_105 (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 formula_107 of discriminant formula_108. 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 formula_71:
Complex multiplication.
Another equivalence for "no Siegel zeros" for formula_102 can be given in terms of for of :
where:
The number formula_116 generates the of formula_72, 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 formula_118 with by formula_119, we have
where formula_121 denotes the . Using the identities formula_122 and formula_123, Colmez' theorem also provides a proof for the equivalence above.
西格尔零点存在所造成的结果.
尽管一般预期广义黎曼猜想是对的，但由于「西格尔零点不存在」的猜想依旧开放之故，因此研究「假如广义黎曼猜想如此的反例存在的话，会有什么结果」，也是一个令人感兴趣的题目。
另一个研究如此可能性的理由，是迄今为止，部分的无条件证明要分成两部分：第一部分是假定西格尔零点不存在，第二部分是假定西格尔零点存在，并证明说想要的定理在这两种状况下都成立。一个如此为之的经典案例是关于算数数列中最小的质数的林尼克定理。
以下是在西格尔零点存在的状况下，所会造成的结果。
存在无限多个孪生质数.
罗杰·希斯-布朗在1983年做出的一个令人惊讶的结果，用陶哲轩的话，可如下陈述：
换句话说，如果(1)不成立，也就是西格尔零点存在的话，那(2)就必须成立；反之若(1)成立，也就是西格尔零点不存在的话，那(2)是否成立依旧是未知数。
筛法的奇偶性问题.
筛法的奇偶性问题指的是筛法无法显示出筛选出的整数有奇数个或偶数个质因数这样的问题。
这使得很多运用筛法的估计，像是使用线性筛（linear sieve）做出的估计，会以一个2的因子，与预期值产生误差。
在2020年，证明说假若西格尔零点存在，那么筛法筛选区间的一般上界就是最佳的，换句话说，在这种状况下，奇偶性多出来的这个2的因子，就不会是筛法的人为限制。