西格爾零點、西格爾零、蘭道 西格爾零點、異常零點（See Iwaniec 2006），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。 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。 The way in which Siegel zeros appear in the theory of Dirichlet Lfunctions is as potential exceptions to the which can only occur when the Lfunction is associated to a real Dirichlet character Real primitive Dirichlet characters For an integer a Dirichlet character modulo is an chimathbb Ztomathbb C satisfying the following properties chi mn chi mchi n for every Periodic chi nq chi n for every Support chi n 0 if mathrm gcdnq 1 That is is the lifting of a widetilde chimathbb Zqmathbb Ztimes to mathbb C The trivial character is the character modulo 1 and the principal character modulo denoted chi_0mathrm modq is the lifting of the trivial homomorphism mathbb Zqmathbb Ztimesni a mapsto 1 in mathbb C A character chimathrm modq is called imprimitive if there exists some integer dneq q with dmid q such that the induced homomorphism widetilde chimathbb Zqmathbb Ztimes to mathbb C factors as mathbb Zqmathbb Ztimestwoheadrightarrow mathbb Zdmathbb Ztimes xrightarrow widetilde chi mathbb C for some character chimathrm modd otherwise chimathrm modq is called primitive A character chi is real or quadratic if it equals its overline chi defined as overline chin overline chi n or equivalently if chi2 chi_0 The real primitive Dirichlet characters are in onetoone correspondence with the 克羅內克符號 s Dcdot mathbb Z to 101 for Dinmathbb Z a ie the discriminant of aSee Satz 4 5 of Zagier 1981 One way to define Dcdot is as the completely multiplicative arithmetic function determined by for prime bigg frac Dpbiggbegin cases 1 ptext splits in mathbb Qsqrt D 1 ptext is inert cdots 0 ptext ramifies cdots end cases quad bigg frac D1bigg text sign of D It is thus common to write chi_D Dcdot which are real primitive characters modulo D Classical zerofree regions The Dirichlet Lfunction associated to a character chimathrm modq is defined as the of the 狄利克雷級數 L schi sum_ngeq 1 chi n ns defined for mathrm Res1 where s is a For chi nonprincipal this continuation is otherwise it has a of prod_pmid q1p1 at as its only singularity For mathrm Res1 Dirichlet Lfunctions can be expanded into an 尤拉乘積 L schi prod_p 1 chi p ps1 from where it follows that L schi has no zeros in this region The is equivalent in a certain sense to L 1itchi neq 0 forall tinmathbb R Moreover via the we can reflect these regions through smapsto 1s to conclude that with the exception of negative integers of same parity as is even if and odd if all the other zeros of L schi must lie inside 0mathrm Res1 This region is called the critical strip and zeros in this region are called nontrivial zeros The classical theorem on zerofree regions Grönwall Landau Titchmarsh states that there exists a n effectively computable real number A0 such that writing ssigma it for the complex variable the function L schi has no zeros in the region sigma 1 frac Alog q t2 if chimathrm modq is nonreal 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 socalled Siegel zero The GRH claims that for every chimathrm modq all the nontrivial zeros of L schi lie on the line mathrm Resfrac 12 定義「西格爾零點」 The definition of Siegel zeros as presented ties it to the constant in the zerofree 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 beta_D denotes the largest real zero of L schi_D then 1beta_D gg frac 1logD The possibility of existence or nonexistence 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 SiegelTatuzawas Theorem and the problem An equivalent formulation of no Siegel zeros that does not reference zeros explicitly is the statement frac LL1chi_D O logD The equivalence can be deduced for example by using the zerofree regions and classical estimates for the number of nontrivial zeros of L schi up to a certain heightSee Chapter 16 of Davenport 1980 LandauSiegel 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 chi_D and chi_D real primitive characters to distinct moduli if beta beta are real zeros of L schi_D L schi_D respectively then minbetabeta 1 frac BlogDD 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 mathbb Qsqrt Dsqrt D 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 varepsilon 0 there is C varepsiloninmathbb R_ such that if beta is a real zero of L schi_D then beta 1 C varepsilonDfrac 38 varepsilon However in the same year in the same issue of the same journal Siegel directly improved this estimate to beta 1 C varepsilonDvarepsilon Both Landaus and Siegels proofs provide no explicit way to calculate C varepsiloninmathbb R_ thus being instances of an SiegelTatuzawa 定理 In 1951 proved an almost effective version of Siegels theorem showing that for any fixed 0 varepsilon frac 1112 if D e1varepsilon then L 1chi_D 0655Dvarepsilon with the possible exception of at most one fundamental discriminant Using the almost effectivity of this result 1973 showed that Eulers 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 zerofree regions since zerofree region estimates enjoy deep connections to the arithmetic of quadratic fields For instance the identity zeta_mathbb Qsqrt Ds zeta s L schi_D can be interpreted as an analytic formulation of see The precise relation between the distribution of zeros near and arithmetic comes from L 1chi_D begin cases dfrac 2 piw_D sqrt D h D text if D 0 5em dfrac logvarepsilon_Dsqrt D h D text if D 0 end cases where h D is the of mathbb Qsqrt D w_D is the number of in mathbb Qsqrt D varepsilon_D is the of mathbb Qsqrt D This way estimates for the largest real zero of L schi_D can be translated into estimates for L 1chi_D via for example the fact that Lsigmachi O log2 q for 1frac 1log q leq sigma leq 1See 11 in Chapter 14 of Davenport 1980 which in turn become estimates for h D 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 zeta_Ks sum_Isubseteq mathfrak O_K mathfrak O_K Is of an Kmathbb Q can be written as a product of Dirichlet LfunctionsTheorem 10525 in Thus if zeta_Ks has a Siegel zero there must be some subfield Fsubseteq K with Fmathbb Q 2 such that zeta_Fs has a Siegel zero While for the nonabelian case zeta_Ks can only be factored into more complicated s the same is true Theorem 1974Lemma 8 in Let Kmathbb Q be a number field of degree There is a constant c n 4 if Kmathbb Q is normal 4n otherwise such that if there is a real beta in the range 1 frac c nlogDelta_K leq beta 1 with zeta_K beta 0 then there is a quadratic subfield Fsubseteq K such that zeta_Fbeta0 Here Delta_K is the of the extension Kmathbb Q No Siegel zeros for D 0 When dealing with quadratic fields the case D0 tends to be elusive due to the behaviour of the fundamental unit Thus it is common to treat the cases D0 and D0 separately Much more is known for the negative discriminant case Lower bounds for h D In 1918 showed that no Siegel zeros for D0 implies that h D gg sqrt DlogD1 see for comparison This can be extended to an equivalence as it is a consequence of Theorem 3 in 2000 text No Siegel zeros for D0 quadiffquad h D gg frac sqrt DlogDsum_abc frac 1a where the summation runs over the ax2 bxy cy2 of discriminant D 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 h D h D gg prod_pmid D bigg 1frac 2sqrt pp1bigg logD Complex multiplication Another equivalence for no Siegel zeros for D0 can be given in terms of for of h j tau_D ll logD where h is the absolute for number fields j is the tau_D Dsqrt D2 The number j tau_D generates the of mathbb Qsqrt D which is its maximal unramified abelian extensionTheorem II41 in This equivalence is a direct consequence of the results in GranvilleStark 2000 and can be seen in C Táfula 2019 A precise relation between heights and values of Lfunctions was obtained by 1993 1998 who showed that for an elliptic curve E_Dmathbb C with by mathbb Ztau_D we have 2 h_mathrm FalE_D frac 12 logD frac LL0chi_D log 2pi where h_mathrm Fal denotes the See the diagram in subsection 06 of Colmez 1993 There is small typo in the upper right corner of this diagram that should instead read 2 h_mathrm FalX frac 12log D Using the identities h_mathrm FalE_D frac 112 h j tau_D O log h j tau_D Proposition 21 Chapter X of and frac LL1chi_D frac LL0chi_D logD log 2pi gammaConsequence of the where is the 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（HeathBrown 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，See Chapter 9 of 會以一個 2 的因子，與預期值產生誤差。 在 2020 年，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 參考