The first breakthrough in dealing with these zeros came from Landau, who showed that there exists an effectively computable constant ${\displaystyle B>0}$  such that, for any ${\textstyle \chi _{D}}$  and ${\textstyle \chi _{D'}}$  real primitive characters to distinct moduli, if ${\textstyle \beta ,\beta '}$  are real zeros of ${\textstyle L(s,\chi _{D}),L(s,\chi _{D'})}$  respectively, then

${\displaystyle \min\{\beta ,\beta '\}<1-{\frac {B}{\log |DD'|}}.}$ 

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 ${\textstyle \mathbb {Q} ({\sqrt {D}},{\sqrt {D'}})}$ . 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 ${\textstyle \varepsilon >0}$ , there is ${\textstyle C(\varepsilon )\in \mathbb {R} _{+}}$  such that, if ${\textstyle \beta }$  is a real zero of ${\textstyle L(s,\chi _{D})}$ , then ${\textstyle \beta <1-C(\varepsilon )|D|^{-{\frac {3}{8}}-\varepsilon }}$ . However, in the same year, in the same issue of the same journal, Siegel^[9] directly improved this estimate to

${\displaystyle \beta <1-C(\varepsilon )|D|^{-\varepsilon }.}$ 

Both Landau's and Siegel's proofs provide no explicit way to calculate ${\textstyle C(\varepsilon )\in \mathbb {R} _{+}}$ , thus being instances of an ineffective result.

Siegel–Tatsuzawa Theorem edit

In 1951, Tikao Tatsuzawa proved an 'almost' effective version of Siegel's theorem,^[10] showing that for any fixed ${\textstyle 0<\varepsilon <{\frac {1}{11.2}}}$ , if ${\textstyle |D|>e^{1/\varepsilon }}$  then

${\displaystyle L(1,\chi _{D})>0.655\varepsilon |D|^{-\varepsilon },}$ 

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]

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 ${\textstyle \zeta _{\mathbb {Q} ({\sqrt {D}})}(s)=\zeta (s)L(s,\chi _{D})}$  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 s = 1 and arithmetic comes from Dirichlet's class number formula:

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

where:

• ${\textstyle h(D)}$  is the ideal class number of ${\textstyle \mathbb {Q} ({\sqrt {D}})}$ ;
• ${\textstyle w_{D}}$  is the number of roots of unity in ${\textstyle \mathbb {Q} ({\sqrt {D}})}$  (D < 0);
• ${\textstyle \varepsilon _{D}}$  is the fundamental unit of ${\textstyle \mathbb {Q} ({\sqrt {D}})}$  (D > 0).

This way, estimates for the largest real zero of ${\textstyle L(s,\chi _{D})}$  can be translated into estimates for ${\textstyle L(1,\chi _{D})}$  (via, for example, the fact that ${\textstyle |L'(\sigma ,\chi )|=O(\log ^{2}q)}$  for ${\textstyle 1-{\frac {1}{\log q}}\leq \sigma \leq 1}$ ),^[13] which in turn become estimates for ${\textstyle h(D)}$ . Classical works in the subject treat these three quantities essentially interchangeably, although the case D > 0 brings additional complications related to the fundamental unit.

Siegel zeros as 'quadratic phenomena' edit

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 ${\textstyle \zeta _{K}(s)=\sum _{I\subseteq {\mathfrak {O}}_{K}}[{\mathfrak {O}}_{K}:I]^{-s}}$  of an abelian number field ${\textstyle K/\mathbb {Q} }$  can be written as a product of Dirichlet L-functions.^[14] Thus, if ${\textstyle \zeta _{K}(s)}$  has a Siegel zero, there must be some subfield ${\textstyle F\subseteq K}$  with ${\textstyle [F:\mathbb {Q} ]=2}$  such that ${\textstyle \zeta _{F}(s)}$  has a Siegel zero.

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

• Theorem (Stark, 1974).^[15] Let ${\textstyle K/\mathbb {Q} }$  be a number field of degree n > 1. There is a constant ${\textstyle c(n)}$  (${\textstyle =4}$  if ${\textstyle K/\mathbb {Q} }$  is normal, ${\textstyle =4n!}$  otherwise) such that, if there is a real ${\textstyle \beta }$  in the range

$1-{\frac {c(n)}{\log |\Delta _{K}|}}\leq \beta <1$ 

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

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

Lower bounds for h(D) edit

In 1918, Erich Hecke showed that "no Siegel zeros" for ${\textstyle D<0}$  implies that ${\textstyle h(D)\gg {\sqrt {|D|}}(\log |D|)^{-1}}$ ^[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]

${\displaystyle {\text{“No Siegel zeros” for }}D<0\quad \iff \quad h(D)\gg {\frac {\sqrt {|D|}}{\log |D|}}\sum _{(a,b,c)}{\frac {1}{a}},}$ 

where the summation runs over the reduced binary quadratic forms ${\textstyle ax^{2}+bxy+cy^{2}}$  of discriminant ${\textstyle D}$ . 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 ${\textstyle h(D)}$ :

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

Complex multiplication edit

Another equivalence for "no Siegel zeros" for ${\textstyle D<0}$  can be given in terms of upper bounds for heights of singular moduli:

${\displaystyle h(j(\tau _{D}))\ll \log |D|,}$ 

where:

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

The number ${\textstyle j(\tau _{D})}$  generates the Hilbert class field of ${\textstyle \mathbb {Q} ({\sqrt {D}})}$ , 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 ${\textstyle E_{D}/\mathbb {C} }$  with complex multiplication by ${\textstyle \mathbb {Z} [\tau _{D}]}$ , we have

${\displaystyle -2h_{\mathrm {Fal} }(E_{D})-{\frac {1}{2}}\log |D|={\frac {L'}{L}}(0,\chi _{D})+\log 2\pi ,}$ 

where ${\textstyle h_{\mathrm {Fal} }}$  denotes the Faltings height.^[22] Using the identities ${\textstyle h_{\mathrm {Fal} }(E_{D})={\frac {1}{12}}h(j(\tau _{D}))+O(\log h(j(\tau _{D})))}$ ^[23] and ${\textstyle {\frac {L'}{L}}(1,\chi _{D})=-{\frac {L'}{L}}(0,\chi _{D})-\log |D|+\log 2\pi +\gamma }$ ,^[24] Colmez' theorem also provides a proof for the equivalence above.

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 edit

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 edit

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.

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

• Davenport, H. (1980). Multiplicative Number Theory. Graduate Texts in Mathematics. Vol. 74. doi:10.1007/978-1-4757-5927-3. ISBN 978-1-4757-5929-7. ISSN 0072-5285.
• Iwaniec, H. (2006), Friedlander, J. B.; Heath-Brown, D. R.; Iwaniec, H.; Kaczorowski, J. (eds.), "Conversations on the Exceptional Character", Analytic Number Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 11–18, 2002, Lecture Notes in Mathematics, vol. 1891, Berlin, Heidelberg: Springer, pp. 97–132, doi:10.1007/978-3-540-36364-4_3, ISBN 978-3-540-36364-4
• Montgomery, H. L.; Vaughan, R. C. (2006). Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. ISBN 978-0-521-84903-6.
• Zagier, D. B. (1981). Zetafunktionen und quadratische Körper: Eine Einführung in die höhere Zahlentheorie. Hochschultext (in German). Berlin Heidelberg: Springer-Verlag. ISBN 978-3-540-10603-6.