西格爾零點

西格爾零點、西格爾零，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函式的零點。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函式有與黎曼ζ函式相似的無零點區域。已隱藏部分未翻譯內容，歡迎參與[ 翻譯]。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 "q" is an formula_1 satisfying the following properties:That is, "χ" is the lifting of a formula_6.The trivial character is the character modulo 1, and the principal character modulo "q", 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 asformula_13for 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 克羅內克符號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 "p" prime):formula_23It 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 狄利克雷級數 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 "s"1 as its only singularity. For formula_28, Dirichlet L-functions can be expanded into an 尤拉乘積 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 "χ", 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 regionformula_42if 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.定義「西格爾零點」.The definition of Siegel zeros as presented ties it to 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: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:formula_51The 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  0 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, thenformula_57This 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 toformula_64Both 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 thenformula_68"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 "s"1 and arithmetic comes from :formula_70where: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  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 , 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:formula_95with formula_96, then there is a quadratic subfield formula_85 such that formula_98. Here, formula_99 is the of the extension formula_83."No Siegel zeros" for "D" &lt; 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):formula_106where 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:formula_110Complex multiplication.Another equivalence for "no Siegel zeros" for formula_102 can be given in terms of for of :formula_112where: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 haveformula_120where formula_121 denotes the . Using the identities formula_122 and formula_123, 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: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.