Siegel zero

E747887

concept in analytic number theory exceptional zero of Dirichlet L-function hypothetical real zero

A Siegel zero is a hypothetical exceptional real zero of certain Dirichlet L-functions that would lie unusually close to 1 and have deep implications for the distribution of prime numbers in arithmetic progressions.

Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Siegel zero problem	1

Statements (49)

Predicate	Object
instanceOf	concept in analytic number theory ⓘ exceptional zero of Dirichlet L-function ⓘ hypothetical real zero ⓘ
affects	Chebotarev density theorem error terms ⓘ distribution of primes in residue classes ⓘ error term in prime number theorem for arithmetic progressions ⓘ
alsoCalled	Landau–Siegel zero NERFINISHED ⓘ
associatedWith	Dirichlet L-function modulo q NERFINISHED ⓘ quadratic Dirichlet character ⓘ real primitive Dirichlet character ⓘ
constrainedBy	Siegel’s theorem on L(1,χ) NERFINISHED ⓘ zero-free regions for Dirichlet L-functions ⓘ
dependsOn	modulus q of the Dirichlet character ⓘ
discussedIn	literature on exceptional zeros of L-functions ⓘ research on primes in arithmetic progressions ⓘ
hasApproximateForm	β with β>1−c/(log q) for some small c>0 ⓘ
hasConsequence	exceptional bias in Chebyshev’s bias ⓘ exceptionally good lower bounds for L(1,χ) ⓘ large deviations from expected distribution of primes mod q ⓘ
hasImplication	ineffective constants in some number theoretic estimates ⓘ ineffectivity in lower bounds for class numbers of quadratic fields ⓘ
hasOpenProblem	existence of Siegel zeros for any modulus ⓘ nonexistence of Siegel zeros for all Dirichlet L-functions ⓘ
hasProperty	real ⓘ simple zero ⓘ
hasRealPart	very close to 1 ⓘ
implies	strong irregularities in distribution of primes in arithmetic progressions ⓘ
isExcludedFor	many small moduli q by explicit computations ⓘ
isHypothetical	true ⓘ
isNear	pole of Riemann zeta function at s=1 ⓘ
isPartOf	study of zeros of L-functions ⓘ theory of Dirichlet L-functions ⓘ
isZeroOf	Dirichlet L-function NERFINISHED ⓘ L(s,χ) for a real Dirichlet character χ ⓘ
liesIn	critical strip of Dirichlet L-function ⓘ
liesOn	real axis ⓘ
mathematicalField	number theory ⓘ
mathematicalSubfield	analytic number theory ⓘ
namedAfter	Carl Ludwig Siegel NERFINISHED ⓘ
relatedTo	Deuring–Heilbronn phenomenon NERFINISHED ⓘ Generalized Riemann Hypothesis NERFINISHED ⓘ Landau–Siegel zeros NERFINISHED ⓘ Siegel’s lower bound for class numbers ⓘ class numbers of imaginary quadratic fields ⓘ
status	not known to be impossible ⓘ unproven to exist ⓘ
studiedIn	analytic theory of L-functions ⓘ
wouldCause	exceptional behavior of class numbers ⓘ
wouldViolate	standard zero-free region near s=1 for Dirichlet L-functions ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Deuring–Heilbronn phenomenon → involves → Siegel zero ⓘ

Carl Ludwig Siegel → notableWork → Siegel zero ⓘ

Deuring–Heilbronn phenomenon → relatedTo → Siegel zero ⓘ

this entity surface form: Siegel zero problem