Siegel zero
Encyclopedia
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 Dirichlet L-function.
There are hypothetical values s of a complex variable, very near (in a quantifiable sense) to 1, such that
for a Dirichlet character
χ, to modulus q say. 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
).
The possibility of a Siegel zero in analytic terms leads to an ineffective estimate
where C is a function of ε for which the proof provides no explicit lower bound (see effective results in number theory
).
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. This was by a 'twisting' argument, implicitly about the L-function of biquadratic field
s. 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 transcendence theory
, and then Dorian Goldfeld's work combined with the Gross-Zagier theorem on Heegner point
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically in the field of analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
, a Siegel zero, named after Carl Ludwig Siegel
Carl Ludwig Siegel
Carl Ludwig Siegel was a mathematician specialising in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.-Biography:...
, is a type of potential counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
to the generalized Riemann hypothesis
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
, on the zeroes of Dirichlet L-function.
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
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
χ, to modulus q say. Important results on this type of zero of an L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
were obtained in the 1930s by Carl Ludwig Siegel
Carl Ludwig Siegel
Carl Ludwig Siegel was a mathematician specialising in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.-Biography:...
, 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
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
).
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
Effective results in number theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable...
).
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
Jacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
); and, that for each modulus q there can be at most one such. This was by a 'twisting' argument, implicitly about the L-function of biquadratic field
Biquadratic field
In mathematics, a biquadratic field is a number field K of a particular kind, which is a Galois extension of the rational number field Q with Galois group the Klein four-group. Such fields are all obtained by adjoining two square roots. Therefore in explicit terms we havefor rational numbers a and b...
s. 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
Kurt Heegner
Kurt Heegner was a German private scholar from Berlin, who specialized inradio engineering and mathematics...
's work, from transcendence theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, and then Dorian Goldfeld's work combined with the Gross-Zagier theorem on Heegner point
Heegner point
In mathematics, a Heegner point is a point on a modular elliptic curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class...
s.