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Potential counterexample to the Generalized Riemann Hypothesis
In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero (also known as 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. Roughly speaking, these are possible zeros very near (in a quantifiable sense) to .

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.

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.

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

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.

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.