Siegel zero explained

In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero, also known as exceptional zero^[1]), named after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions associated to quadratic number fields. Roughly speaking, these are possible zeros very near (in a quantifiable sense) to

s =1 . (The pronunciation of "Siegel" begins with a Z sound.)

Motivation and definition

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, a Dirichlet character modulo is an arithmetic function $\backslash chi\backslash colon\; \backslash mathbb\backslash to\backslash mathbb$ satisfying the following properties:

• (Completely multiplicative) $\backslash chi(mn)\; =\; \backslash chi(m)\backslash chi(n)$ for every, ;
• (Periodic) $\backslash chi(n+q)\; =\; \backslash chi(n)$ for every ;
• (Support) $\backslash chi(n)\; =\; 0$ if and only if

gcd (n,q) > 1 .That is, is the lifting of a homomorphism $\backslash widetilde:(\backslash mathbb/q\backslash mathbb)^\; \backslash to\; \backslash mathbb^$.

The trivial character is the character modulo 1, and the principal character modulo, denoted $\backslash chi\_0~(\backslash mathrm~q)$, is the lifting of the trivial homomorphism $(\backslash mathbb/q\backslash mathbb)^\backslash ni\; a\; \backslash mapsto\; 1\; \backslash in\; \backslash mathbb^$.

A character $\backslash chi~(\backslash mathrm~)$ is called imprimitive if there exists some integer $d\backslash neq\; q$ with $d\backslash mid\; q$ such that the induced homomorphism $\backslash widetilde\backslash colon\; (\backslash mathbb/q\backslash mathbb)^\; \backslash to\; \backslash mathbb^$ factors as

( Z /q Z ) ^{x} \twoheadrightarrow ( Z /d Z ) ^{x} \xrightarrow{\widetilde{\chi'}} C ^{*} for some character $\backslash chi\text{'}~(\backslash mathrm~)$; otherwise, $\backslash chi~(\backslash mathrm~)$ is called primitive.\overline{\chi} (defined as \overline{\chi}(n) : = \overline{\chi(n)} ), or equivalently if $\backslash chi^2\; =\; \backslash chi\_0$. The real primitive Dirichlet characters are in one-to-one correspondence with the Kronecker symbols $(D|\backslash ,\backslash cdot\backslash ,):\; \backslash mathbb\; \backslash to\; \backslash$ for $D\backslash in\backslash mathbb$ a fundamental discriminant (i.e., the discriminant of a quadratic number field).^[2] One way to define $(D|\backslash ,\backslash cdot\backslash ,)$ is as the completely multiplicative arithmetic function determined by (for prime):( (D)/(p) ) = \begin{cases} 1 , &(p) splitsin Q (\sqrt{D}), \ -1 , &(p) isinert … , \ 0 , &(p) ramifies … , \end{cases} ( (D)/(-1) ) = signof D. It is thus common to write $\backslash chi\_D\; :=\; (D|\backslash ,\backslash cdot\backslash ,)$, which are real primitive characters modulo $|D|$.

Classical zero-free regions

The Dirichlet L-function associated to a character $\backslash chi~(\backslash mathrm~q)$ is defined as the analytic continuation of the Dirichlet series $L(s,\backslash chi)\; =\; \backslash sum\_\; \backslash chi(n)\; n^$ defined for $\backslash mathrm(s)>1$, where s is a complex variable. For

\chi non-principal, this continuation is entire; otherwise it has a simple pole of residue $\backslash prod\_(1-p^)$ at as its only singularity. For $\backslash mathrm(s)>1$, Dirichlet L-functions can be expanded into an Euler product $L(s,\backslash chi)\; =\; \backslash prod\_\; (1\; -\; \backslash chi(p)p^)^$, from where it follows that $L(s,\backslash chi)$ has no zeros in this region. The prime number theorem for arithmetic progressions is equivalent (in a certain sense) to $L(1+it,\backslash chi)\; \backslash neq\; 0$ ($\backslash forall\; t\backslash in\backslash mathbb$). Moreover, via the functional equation, we can reflect these regions through $s\backslash mapsto\; 1-s$ to conclude that, with the exception of negative integers of same parity as,^[3] all the other zeros of $L(s,\backslash chi)$ must lie inside \{ 0 < Re (s)< 1 \} . 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,^[4] Landau,^[5] Titchmarsh^[6]) states that there exists an effectively computable real number $A>0$ such that, writing

s = \sigma + it for the complex variable, the function $L(s,\backslash chi)$ has no zeros in the region\sigma > 1 - (A)/(( log q(|t| +2 ))) if $\backslash chi~(\backslash mathrm~q)$ is non-real. If $\backslash chi$ is real, then there is at most one zero in this region, which must necessarily be real and simple. This possible zero is the so-called Siegel zero.

The Generalized Riemann Hypothesis (GRH) claims that for every $\backslash chi~(\backslash mathrm~q)$, all the non-trivial zeros of $L(s,\backslash chi)$ lie on the line $\backslash mathrm(s)=\backslash frac$.

Defining "Siegel zeros"

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 $\backslash beta\_D$ denotes the largest real zero of $L(s,\backslash chi\_D)$, then

1- \beta _{D} \gg (1)/(log |D|) .

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–Tatsuzawa's Theorem and the idoneal number problem). An equivalent formulation of "no Siegel zeros" that does not reference zeros explicitly is the statement:

(L')/(L) ( 1 ,\chi _{D)} = O( log |D|). The equivalence can be deduced for example by using the zero-free regions and classical estimates for the number of non-trivial zeros of $L(s,\backslash chi)$ up to a certain height.^[7]

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 $\backslash chi\_D$ and $\backslash chi\_$ real primitive characters to distinct moduli, if $\backslash beta,\; \backslash beta\text{'}$ are real zeros of $L(s,\backslash chi\_D),\; L(s,\backslash chi\_)$ respectively, thenmin \{\beta,\beta'\} < 1- (B)/(log |DD'|) . 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 $\backslash mathbb(\backslash sqrt,\backslash sqrt)$. 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,^[8] which states that for every $\backslash varepsilon\; >\; 0$, there is $C(\backslash varepsilon)\backslash in\backslash mathbb\_$ such that, if $\backslash beta$ is a real zero of $L(s,\backslash chi\_D)$, then . However, in the same year, in the same issue of the same journal, Siegel^[9] directly improved this estimate to

\beta < 1 - C(\varepsilon)|D| ^{-\varepsilon} . Both Landau's and Siegel's proofs provide no explicit way to calculate $C(\backslash varepsilon)\backslash in\backslash mathbb\_$, thus being instances of an ineffective result.

Siegel–Tatsuzawa Theorem

In 1951, Tikao Tatsuzawa proved an 'almost' effective version of Siegel's theorem,^[10] showing that for any fixed , if $|D|\; >\; e^$ then

L( 1 ,\chi _{D)} > 0 . 655 \varepsilon|D| ^{-\varepsilon} , with the possible exception of at most one fundamental discriminant. Using the 'almost effectivity' of this result, P. J. Weinberger (1973)^[11] showed that Euler's list of 65 idoneal numbers is complete except for at most two elements.^[12]

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 $\backslash zeta\_(s)\; =\; \backslash zeta(s)\; L(s,\backslash chi\_D)$ 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:

L( 1 ,\chi _{D)} = \begin{cases} \dfrac{ 2 \pi}{w _{D} \sqrt{|D|}} h(D), & if D < 0 \\[. 5 em] \dfrac{ log \varepsilon _{D}{\sqrt{D}}} h(D), & if D > 0 , \end{cases} where:

• $h(D)$ is the ideal class number of $\backslash mathbb(\backslash sqrt)$;
• $w\_D$ is the number of roots of unity in $\backslash mathbb(\backslash sqrt)$ ;
• $\backslash varepsilon\_D$ is the fundamental unit of $\backslash mathbb(\backslash sqrt)$ .

This way, estimates for the largest real zero of $L(s,\backslash chi\_D)$ can be translated into estimates for $L(1,\backslash chi\_D)$ (via, for example, the fact that $|L\text{'}(\backslash sigma,\backslash chi)|\; =\; O(\backslash log^2\; q)$ for $1-\backslash frac\; \backslash leq\; \backslash sigma\; \backslash leq\; 1$),^[13] 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 Kronecker–Weber theorem, for example, that the Dedekind zeta function $\backslash zeta\_(s)\; =\; \backslash sum\_\; [\backslash mathfrak\{O\}\_K:\; I]^$ of an abelian number field $K/\backslash mathbb$ can be written as a product of Dirichlet L-functions.^[14] Thus, if $\backslash zeta\_(s)$ has a Siegel zero, there must be some subfield $F\backslash subseteq\; K$ with $[F:\backslash mathbb\{Q\}]\; =\; 2$ such that $\backslash zeta\_(s)$ has a Siegel zero.

While for the non-abelian case $\backslash zeta\_(s)$ can only be factored into more complicated Artin L-functions, the same is true:

• Theorem (Stark, 1974).^[15] Let $K/\backslash mathbb$ be a number field of degree . There is a constant $c(n)$ ($=\; 4$ if $K/\backslash mathbb$ is normal, $=\; 4n!$ otherwise) such that, if there is a real $\backslash beta$ in the range

with $\backslash zeta\_K(\backslash beta)\; =\; 0$, then there is a quadratic subfield $F\backslash subseteq\; K$ such that $\backslash zeta\_(\backslash beta)=0$. Here, $\backslash Delta\_K$ is the field discriminant of the extension $K/\backslash mathbb$.

"No Siegel zeros" for D < 0

When dealing with quadratic fields, the case $D>0$ tends to be elusive due to the behaviour of the fundamental unit. Thus, it is common to treat the cases and $D>0$ separately. Much more is known for the negative discriminant case:

Lower bounds for h(D)

In 1918, Erich Hecke showed that "no Siegel zeros" for implies that $h(D)\; \backslash gg\; \backslash sqrt$

(\log|D|)^^[5] (see Class number problem for comparison). This can be extended to an equivalence, as it is a consequence of Theorem 3 in Granville–Stark (2000):^[16] “NoSiegelzeros”for D< 0 \iff h(D) \gg (\sqrt{|D|)/() }\sum_ \frac, where the summation runs over the reduced binary quadratic forms $ax^2\; +\; bxy\; +\; cy^2$ of discriminant $D$. Using this, Granville and Stark showed that a certain uniform formulation of the abc conjecture for number fields implies "no Siegel zeros" for negative discriminants.

In 1976, Dorian Goldfeld^[17] proved the following unconditional, effective lower bound for $h(D)$:

h(D) \gg \prod _{p\mid} ( 1- (2 \sqrt{p)/() }\bigg)\, \log|D|.

Complex multiplication

Another equivalence for "no Siegel zeros" for can be given in terms of upper bounds for heights of singular moduli:

h(j(\tau _{D))} \ll log |D|, where:

• $h$ is the absolute logarithmic naïve height for number fields;
• $j$ is the j-invariant function;
• $\backslash tau\_D\; :=\; (D+\backslash sqrt)/2$.

The number $j(\backslash tau\_D)$ generates the Hilbert class field of $\backslash mathbb(\backslash sqrt)$, which is its maximal unramified abelian extension.^[18] This equivalence is a direct consequence of the results in Granville–Stark (2000),^[16] and can be seen in C. Táfula (2019).^[19]

A precise relation between heights and values of L-functions was obtained by Pierre Colmez (1993,^[20] 1998^[21]), who showed that, for an elliptic curve $E\_D/\backslash mathbb$ with complex multiplication by $\backslash mathbb[\backslash tau\_D]$, we have

-2 h _{Fal} (E _{D)} - (1)/(2) log |D| = (L')/(L) ( 0 ,\chi _{D)} + log 2 \pi, where $h\_$ denotes the Faltings height.^[22] Using the identities $h\_(E\_D)\; =\; \backslash frach(j(\backslash tau\_D))\; +\; O(\backslash log\; h(j(\backslash tau\_D)))$^[23] and $\backslash frac(1,\backslash chi\_D)\; =\; -\backslash frac(0,\backslash chi\_D)\; -\; \backslash log|D|\; +\; \backslash log\; 2\backslash pi\; +\; \backslash gamma$,^[24] Colmez' theorem also provides a proof for the equivalence above.

Consequences of Siegel zeros existing

Although the Generalized Riemann Hypothesis 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 Linnik's theorem on the smallest prime in an arithmetic progression.

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

Infinitude of twin primes

See main article: Twin prime.

A striking result in this direction is Roger Heath-Brown's 1983 result^[25] which, following Terence Tao,^[26] 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 twin primes.

Parity problem

See main article: Parity problem (sieve theory).

The parity problem in sieve theory 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 linear sieve^[27] being off by a factor of 2 from the expected value. In 2020, Granville^[28] 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.

See also

• Generalized Riemann hypothesis
• Deuring–Heilbronn phenomenon
• Class number problem
• Brauer–Siegel theorem
• Siegel–Walfisz theorem

References

1. See Iwaniec (2006).
2. See Satz 4, §5 of Zagier (1981).
3. is even if, and odd if .
4. Grönwall . T. H. . Thomas Hakon Grönwall. 1913 . Rendiconti di Palermo . 35 . 145–159 . Sur les séries de Dirichlet correspondant à des charactères complexes . 10.1007/BF03015596 . 121161132 . fr.
5. . E. . 1918 . Göttinger Nachrichten . 285–295 . Über die Klassenzahl imaginär-quadratischer Zahlkörper . de.
6. Titchmarsh . E. C. . Edward Charles Titchmarsh. 1930 . Rendiconti di Palermo . 54 . 414–429 . A divisor problem. 10.1007/BF03021203 . 119578445 .
7. See Chapter 16 of Davenport (1980).
8. Landau . E. . Edmund Landau. 1936 . . 1–18 . Bemerkungen zum Heilbronnschen Satz . de.
9. Siegel . C. L. . 1935 . . 1 . 1 . 83–86 . Über die Klassenzahl quadratischer Zahlkörper . 10.4064/aa-1-1-83-86 . de . On the class numbers of quadratic fields. free .
10. Tatuzawa . T. . 1951 . Japanese Journal of Mathematics . 21 . 163–178 . On a theorem of Siegel. 10.4099/jjm1924.21.0_163 . free .
11. Weinberger . P. J. . 1973 . Acta Arithmetica . 22 . 117–124 . Exponents of the class group of complex quadratic fields. 2 . 10.4064/aa-22-2-117-124 . free .
12. Kani . Ernst . 2011 . Idoneal numbers and some generalizations . Annales des Sciences Mathématiques du Québec . 35 . 2 . Remark 24.
13. See (11) in Chapter 14 of Davenport (1980).
14. Theorem 10.5.25 in Book: Cohen, H.. Henri Cohen (number theorist). Number Theory: Volume II: Analytic and Modern Tools. 2007. Springer-Verlag. 978-0-387-49893-5. Graduate Texts in Mathematics, Number Theory. New York. en. .
15. Lemma 8 in Stark. H. M.. 1974-06-01. Some effective cases of the Brauer-Siegel Theorem. Inventiones Mathematicae. en. 23. 2. 135–152. 10.1007/BF01405166. 1974InMat..23..135S . 119482000. 1432-1297.
16. Granville. A.. Stark. H.M.. 2000-03-01. ABC implies no "Siegel zeros" for L-functions of characters with negative discriminant. Inventiones Mathematicae. en. 139. 3. 509–523. 10.1007/s002229900036. 2000InMat.139..509G . 6901166. 1432-1297.
17. Goldfeld. Dorian M.. Dorian Goldfeld. 1976. The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. fr. 3. 4. 623–663.
18. Theorem II.4.1 in .
19. Táfula, C.. On Landau–Siegel zeros and heights of singular moduli. Acta Arithmetica. 2021. 201. 1–28. 10.4064/aa191118-18-5. 1911.07215. 208138549.
20. Colmez. Pierre. Pierre Colmez. 1993. Periodes des Variétés Abéliennes à Multiplication Complexe. Annals of Mathematics. 138. 3. 625–683. 10.2307/2946559. 2946559. 0003-486X.
21. Colmez. Pierre. 1998-05-01. Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe. Compositio Mathematica. en. 111. 3. 359–369. 10.1023/A:1000390105495. 1570-5846. free.
22. See the diagram in subsection 0.6 of Colmez (1993). There is small typo in the upper right corner of this diagram, that should instead read "$-2\; h\_(X)\; -\; \backslash frac\backslash log\; D$".
23. Proposition 2.1, Chapter X of Book: Arithmetic Geometry. 1986. Springer-Verlag. 978-0-387-96311-2. Cornell. G.. New York. en. Silverman. J. H..
24. Consequence of the functional equation, where is the Euler–Mascheroni constant.
25. Heath-Brown. D. R.. Roger Heath-Brown. 1983-09-01. Prime Twins and Siegel Zeros. Proceedings of the London Mathematical Society. en. s3-47. 2. 193–224. 10.1112/plms/s3-47.2.193. 0024-6115.
26. Web site: 2015-08-27. Heath-Brown's theorem on prime twins and Siegel zeroes. 2021-03-13. What's new. en.
27. See Chapter 9 of Book: Nathanson, Melvyn B.. Additive Number Theory The Classical Bases. 1996. Springer-Verlag. 978-0-387-94656-6. Graduate Texts in Mathematics. New York. en.
28. Granville, A.. Sieving intervals and Siegel zeros. 2020. math.NT. 2010.01211.

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