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.
satisfying the following properties: That is, χ is the lifting of a homomorphism
The real primitive Dirichlet characters are in one-to-one correspondence with the Kronecker symbols
is as the completely multiplicative arithmetic function determined by (for p prime): It is thus common to write
non-principal, this continuation is entire; otherwise it has a simple pole of residue
The prime number theorem for arithmetic progressions is equivalent (in a certain sense) to
to conclude that, with the exception of negative integers of same parity as χ,[3] all the other zeros of
The classical theorem on zero-free regions (Grönwall,[4] Landau,[5] Titchmarsh[6]) states that there exists an effectively computable real number
The definition of Siegel zeros as presented ties it to the constant A 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 A is of little concern.
[1] 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–Tatsuzawa's Theorem and the idoneal number problem).
[7] The first breakthrough in dealing with these zeros came from Landau, who showed that there exists an effectively computable constant
The way this is proved is via a 'twisting' argument, which lifts the problem to the Dedekind zeta function of the biquadratic field
This 'repelling effect' (see Deuring–Heilbronn phenomenon), after more careful analysis, led Landau to his 1936 theorem,[8] which states that for every
In 1951, Tikao Tatsuzawa proved an 'almost' effective version of Siegel's theorem,[10] showing that for any fixed
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.
Classical works in the subject treat these three quantities essentially interchangeably, although the case D > 0 brings additional complications related to the fundamental unit.
There is a sense in which the difficulty associated to the phenomenon of Siegel zeros in general is entirely restricted to quadratic extensions.
can only be factored into more complicated Artin L-functions, the same is true: When dealing with quadratic fields, the case
Much more is known for the negative discriminant case: In 1918, Erich Hecke showed that "no Siegel zeros" for
This can be extended to an equivalence, as it is a consequence of Theorem 3 in Granville–Stark (2000):[16] 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[17] proved the following unconditional, effective lower bound for
can be given in terms of upper bounds for heights of singular moduli: where: The number
[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
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 striking result in this direction is Roger Heath-Brown's 1983 result[25] which, following Terence Tao,[26] can be stated as follows: 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.
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.