Siegel’s theorem on zeros of L-functions

E747888

result about Dirichlet L-functions theorem in analytic number theory

Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.

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Statements (47)

Predicate	Object
instanceOf	result about Dirichlet L-functions ⓘ theorem in analytic number theory ⓘ
appliesTo	Dirichlet L-functions NERFINISHED ⓘ L-functions associated to Dirichlet characters ⓘ
assumes	Dirichlet characters modulo q ⓘ
concerns	nontrivial zeros of L-functions ⓘ zeros near 1 of Dirichlet L-functions ⓘ zeros of Dirichlet L-functions ⓘ
context	classical theory of Dirichlet L-series ⓘ study of the distribution of primes in residue classes ⓘ
describes	exceptional real zeros of L(s, χ) close to 1 ⓘ location of real zeros of L(s, χ) with χ real ⓘ
feature	ineffectivity of the implied constants ⓘ possible existence of an exceptional real zero (Siegel zero) ⓘ
field	analytic number theory ⓘ
gives	bounds on how close nontrivial zeros can approach 1 ⓘ lower bounds for 1 − β where β is a real zero of a Dirichlet L-function ⓘ
hasAlternativeName	Siegel’s theorem on exceptional zeros NERFINISHED ⓘ Siegel’s theorem on real zeros of Dirichlet L-functions NERFINISHED ⓘ
hasConsequence	effective bounds for primes in arithmetic progressions up to a possible exceptional modulus ⓘ ineffective constants in some prime distribution estimates ⓘ strong error terms in the prime number theorem for arithmetic progressions ⓘ
hasProperty	non-effective (ineffective) nature of the constant in the bound ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	restrictions on Siegel zeros ⓘ strong bounds on exceptional real zeros of Dirichlet L-functions ⓘ zeros of Dirichlet L-functions cannot lie too close to 1 ⓘ
influenced	later work on zero-free regions for L-functions ⓘ research on eliminating or controlling Siegel zeros ⓘ
involves	Dirichlet L-function L(s, χ) NERFINISHED ⓘ real characters modulo q ⓘ zeros on the real axis close to s = 1 ⓘ
isAbout	how close a real zero of L(s, χ) can be to 1 in terms of the modulus q ⓘ
mathematicalDomain	complex analysis ⓘ number theory ⓘ
namedAfter	Carl Ludwig Siegel NERFINISHED ⓘ
provedBy	Carl Ludwig Siegel NERFINISHED ⓘ
provides	ineffective lower bounds for 1 − β in terms of the modulus q ⓘ
relatedTo	Dirichlet’s theorem on arithmetic progressions NERFINISHED ⓘ Generalized Riemann Hypothesis NERFINISHED ⓘ Landau–Page theorem NERFINISHED ⓘ Siegel zero ⓘ prime number theorem for arithmetic progressions ⓘ zero-free regions for L-functions ⓘ
usedFor	distribution of primes in arithmetic progressions ⓘ
usedIn	bounding error terms in Chebotarev-type results for abelian extensions ⓘ proofs of strong versions of the prime number theorem in arithmetic progressions ⓘ

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Deuring–Heilbronn phenomenon → relatedTo → Siegel’s theorem on zeros of L-functions ⓘ