Math 2.0

Number theory Definition

Siegel zero

Builds on Dirichlet characterDirichlet L-functionGeneralized Riemann hypothesis

Definition

There is an absolute constant c>0c > 0 with the following property. Let q≥3q \ge 3 be an integer and χ\chi a Dirichlet character modulo qq. Then the Dirichlet L-function L(s,χ)L(s, \chi) has no zero s=σ+its = \sigma + it in the region σ≥1−clog⁡(q(∣t∣+2)).\sigma \ge 1 - \frac{c}{\log\big(q(\abs{t} + 2)\big)}. The one possible exception is a single zero when χ\chi is a real nonprincipal character; such a zero is real and simple. Fix such a constant cc. A Siegel zero (or exceptional zero) for the modulus qq is a real number β\beta with 1−clog⁡q<β<1andL(β,χ)=01 - \frac{c}{\log q} < \beta < 1 \quad\text{and}\quad L(\beta, \chi) = 0 for some real nonprincipal character χ\chi modulo qq.

Remarks

No Siegel zero is known, so there is no example to check. If the generalized Riemann hypothesis holds, every zero of L(s,χ)L(s, \chi) in the strip 0<Re⁡s<10 < \operatorname{Re} s < 1 has real part 1/21/2, so for large qq no Siegel zero exists. A non-example: a zero of L(s,χ)L(s, \chi) for a character χ\chi that is not real is never a Siegel zero, because the zero-free region has no exception for such χ\chi. Siegel's theorem bounds a possible exceptional zero: for every ε>0\varepsilon > 0 there is c(ε)>0c(\varepsilon) > 0 with β<1−c(ε)q−ε\beta < 1 - c(\varepsilon) q^{-\varepsilon}, but c(ε)c(\varepsilon) is not effectively computable. The notion depends on the choice of cc; statements about Siegel zeros are usually about a sequence of moduli qq with (1−β)log⁡q→0(1 - \beta)\log q \to 0.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptDirichlet L-function, concept, no evidence levelDirichlet L-functionconceptDirichlet character, concept, no evidence levelDirichlet characterconceptSiegel zero, concept, no evidence levelSiegel zeroconceptZero of a holomorphic function, concept, no evidence levelZero of a holomorphic…conceptGeneralized Riemann hypothesis, conjecture, no evidence levelGeneralized Riemann h…conjectureSiegel zero, definition, no evidence levelSiegel zerodefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Dedekind zeta function concept
  3. Dirichlet L-function concept
  4. Dirichlet character concept
  5. Siegel zero concept
  6. Zero of a holomorphic function concept
  7. Generalized Riemann hypothesis conjecture
  8. Siegel zero definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Siegel_zero
About this page
TagSHFJH5JZ5P
URIm2://SHFJH5JZ5P
Kinddefinition
WritingW0 stub
MSC11M20, 11M26
DefinesSiegel zero
UsesDirichlet character, Dirichlet L-function, Generalized Riemann hypothesis
Changed2 November 2026

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