Siegel zero

In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of a Dirichlet L-function.

Definition

There are hypothetical values s of a complex variable, very near (in a quantifiable sense) to 1, such that

L(s,χ) = 0

for a Dirichlet character χ of modulus q say.

Immediate consequence

The possibility of a Siegel zero in analytic terms leads to an ineffective estimate

L(1,χ) > C(ε)q^−ε

where C is a function of ε for which the proof provides no explicit lower bound (see effective results in number theory).

History

Important results on this type of zero of an L-function were obtained in the 1930s by Carl Ludwig Siegel, from whom they take their name (he was not the first to consider them, and they are sometimes called Landau–Siegel zeroes to acknowledge also the work of Edmund Landau).^[1]

Importance

The importance of the possible Siegel zeroes is seen in all known results on the zero-free regions of L-functions: they show a kind of 'indentation' near s = 1, while otherwise generally resembling that for the Riemann zeta function — that is, they are to the left of the line Re(s) = 1, and asymptotic to it. Because of the analytic class number formula, data on Siegel zeroes have a direct impact on the class number problem, of giving lower bounds for class numbers. This question goes back to C. F. Gauss. What Siegel showed was that such zeroes are of a particular type (namely, that they can occur only for χ a real character, which must be a Jacobi symbol); and, that for each modulus q there can be at most one such.^[2] This was by a 'twisting' argument, implicitly about the L-function of biquadratic fields. This in a sense isolated the Siegel zero as a special case of GRH (which would prove that it did not exist). In subsequent developments, however, detailed information on the Siegel zero has not shown it to be impossible. Work on the class number problem has instead been progressing by methods from Kurt Heegner's work, from algebraic number theory, and then Dorian Goldfeld's work combined with the Gross-Zagier theorem on Heegner points.

References

^ Táfula, Christian (2019). "On Landau-Siegel zeros and heights of singular moduli". arXiv:1911.07215 [math.NT]. (See pages 12–13.)
^ Broughan, Kevin (2 November 2017). Equivalents of the Riemann Hypothesis: Volume 2, Analytic Equivalents. Cambridge University Press. p. 291. ISBN 978-1-108-18702-2.

Siegel, Carl Ludwig (1935). "Über die Classenzahl quadratischer Zahlkörper" [On the class numbers of quadratic fields]. Acta Arithmetica (in German). 1 (1): 83–86. doi:10.4064/aa-1-1-83-86.

[1] Táfula, Christian (2019). "On Landau-Siegel zeros and heights of singular moduli". arXiv:1911.07215 [math.NT]. (See pages 12–13.)

[Broughan2017-2] Broughan, Kevin (2 November 2017). Equivalents of the Riemann Hypothesis: Volume 2, Analytic Equivalents. Cambridge University Press. p. 291. ISBN 978-1-108-18702-2.

[1]