Linnik’s theorem on the least prime in an arithmetic progression

E747889

result in number theory theorem in analytic number theory

Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.

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Statements (45)

Predicate	Object
instanceOf	result in number theory ⓘ theorem in analytic number theory ⓘ
assertsExistenceOf	absolute constant L > 0 independent of q and a ⓘ
concerns	distribution of primes in arithmetic progressions ⓘ least prime in an arithmetic progression ⓘ upper bounds for least primes in residue classes ⓘ
conclusionInvolves	least prime p ≡ a (mod q) ⓘ
field	analytic number theory ⓘ number theory ⓘ
gives	explicit upper bound for the least prime in a coprime residue class modulo q ⓘ
hasConsequence	effective version of Dirichlet’s theorem with explicit dependence on q ⓘ existence of primes in short intervals within arithmetic progressions up to C q^L ⓘ
hasParameter	Linnik exponent L ⓘ modulus q ⓘ residue class a modulo q ⓘ
hasRefinementsBy	Heath-Brown NERFINISHED ⓘ Xylouris NERFINISHED ⓘ other analytic number theorists ⓘ
implies	every coprime residue class modulo q contains a prime not exceeding C q^L ⓘ
improvesOn	trivial exponential bounds for least primes in arithmetic progressions ⓘ
involves	Linnik exponent ⓘ
isDiscussedIn	monographs on analytic number theory ⓘ research articles on least primes in arithmetic progressions ⓘ
isRelatedTo	Chebotarev density theorem NERFINISHED ⓘ Dirichlet’s theorem on arithmetic progressions NERFINISHED ⓘ Linnik’s dispersion method NERFINISHED ⓘ least prime in a given residue class problem ⓘ zero-free regions of Dirichlet L-functions ⓘ
isUsedIn	applications to computational number theory involving prime search in residue classes ⓘ studies of primes in arithmetic progressions with large moduli ⓘ
isWeakerThan	Generalized Riemann Hypothesis bounds for least primes in arithmetic progressions ⓘ
namedAfter	Yuri Vladimirovich Linnik NERFINISHED ⓘ
originallyProvedBy	Yuri Linnik NERFINISHED ⓘ
originalProofUsed	Linnik’s dispersion method NERFINISHED ⓘ
quantifiesOver	all integers a with gcd(a,q)=1 ⓘ all integers q ≥ 1 ⓘ
requiresCondition	residue class a is coprime to modulus q ⓘ
statesThat	there exists a constant L such that the least prime p ≡ a (mod q) with (a,q)=1 satisfies p ≤ C q^L for some constant C ⓘ
strengthens	Dirichlet’s theorem by giving an explicit upper bound for the least prime NERFINISHED ⓘ
subfield	multiplicative number theory ⓘ
typeOfBound	polynomial bound in q ⓘ
usesMethod	analytic techniques involving L-functions ⓘ zero-density estimates for Dirichlet L-functions ⓘ zero-free regions for Dirichlet L-functions ⓘ
yearProved	1944 ⓘ

Referenced by (1)

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Deuring–Heilbronn phenomenon → relatedTo → Linnik’s theorem on the least prime in an arithmetic progression ⓘ