In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero, also known as an exceptional zero, 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.

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 satisfying the following properties:

• Completely multiplicative for every , ;

• Periodic for every ;

• .

That is, is the lifting of a homomorphism.

The trivial character is the character modulo 1, and the principal character modulo .

is called imprimitive factors as

is called primitive.

is real (or quadratic) if it equals its complex conjugate. The real primitive Dirichlet characters are in one-to-one correspondence with the Kronecker symbols a fundamental discriminant (i.e., the discriminant of a quadratic number field). is as the completely multiplicative arithmetic function determined by (for prime):

.

Classical zero-free regions

The Dirichlet L-function is defined as the analytic continuation of the Dirichlet series, where s is a complex variable non-principal, this continuation is entire; otherwise it has a simple pole of residue at , Dirichlet L-functions can be expanded into an Euler product has no zeros in this region. The prime number theorem for arithmetic progressions). Moreover, via the functional equation to conclude that, with the exception of negative integers of same parity as ,. 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 has no zeros in the region

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.

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"):

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:

up to a certain height.

Landau–Siegel estimates

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 Dedekind zeta function of the biquadratic field. 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,. However, in the same year, in the same issue of the same journal, Siegel directly improved this estimate to

, thus being instances of an ineffective result.

Siegel–Tatsuzawa theorem

In 1951, Tikao Tatsuzawa proved an 'almost' effective version of Siegel's theorem, then

with the possible exception of at most one fundamental discriminant. Using the 'almost effectivity' of this result, P. J. Weinberger (1973) showed that Euler's list of 65 idoneal numbers is complete except for at most two elements.

Relation to quadratic fields

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:

where:

• is the ideal class number;

• is the number of roots of unity ();

• is the fundamental unit ().

),. 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 of an abelian number field can be written as a product of Dirichlet L-functions. has a Siegel zero.

can only be factored into more complicated Artin L-functions, the same is true:

• Theorem (Stark, 1974). be a number field of degree in the range

• is the field discriminant.

"No Siegel zeros" for D < 0

separately. Much more is known for the negative discriminant case:

Lower bounds for h(D)

In 1918, Erich Hecke (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):

where the summation runs over the reduced binary quadratic forms. 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:

Complex multiplication

can be given in terms of upper bounds for heights of singular moduli:

where:

• is the absolute logarithmic naïve height for number fields;

• is the j-invariant function;

• .

generates the Hilbert class field, 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 Pierre Colmez (1993, 1998 with complex multiplication, we have

denotes the Faltings height., 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

A striking result in this direction is Roger Heath-Brown's 1983 result which, following Terence Tao, 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

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 being off by a factor of 2 from the expected value. In 2020, Granville 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