Dirichlet's theorem on arithmetic progressions

E466245

result in analytic number theory theorem in number theory

Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.

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Observed surface forms (2)

Surface form	Occurrences
Dirichlet’s theorem on arithmetic progressions	3
Dirichlet’s theorem on primes in arithmetic progressions	1

Statements (46)

Predicate	Object
instanceOf	result in analytic number theory ⓘ theorem in number theory ⓘ
appliesTo	any modulus d ≥ 1 ⓘ any residue class a modulo d with gcd(a,d)=1 ⓘ
assumption	gcd(a,d)=1 for the arithmetic progression a+nd ⓘ
classification	deep theorem of classical number theory ⓘ
condition	The first term and common difference of the arithmetic progression must be coprime ⓘ
consequence	Distribution of primes among reduced residue classes modulo d is infinite in each class ⓘ No congruence class coprime to the modulus can contain only finitely many primes ⓘ
doesNotApplyTo	arithmetic progressions where gcd(a,d) > 1 ⓘ
domain	arithmetic progressions of integers ⓘ
ensures	For any modulus d and residue class a coprime to d, there are infinitely many primes p with p ≡ a (mod d) ⓘ
field	analytic number theory ⓘ number theory ⓘ
generalizes	Euclid's theorem on the infinitude of primes NERFINISHED ⓘ
historicalSignificance	Introduced Dirichlet characters and L-series into number theory ⓘ One of the first major uses of analytic methods in number theory ⓘ
implies	There are infinitely many primes congruent to a modulo d for any a coprime to d ⓘ
importance	fundamental theorem in multiplicative number theory ⓘ
methodOfProof	analytic methods ⓘ use of L-series and complex analysis ⓘ
namedAfter	Johann Peter Dirichlet NERFINISHED ⓘ
provedBy	Johann Peter Dirichlet NERFINISHED ⓘ
relatedConcept	Euler product formula NERFINISHED ⓘ distribution of prime numbers ⓘ non-vanishing of L-functions on the line Re(s)=1 ⓘ reduced residue system modulo n ⓘ
relatedTo	Chebotarev density theorem NERFINISHED ⓘ Dirichlet L-function NERFINISHED ⓘ Dirichlet character NERFINISHED ⓘ prime number theorem for arithmetic progressions ⓘ
specialCase	Infinitude of primes in the progression -1 mod n for any n with gcd(-1,n)=1 ⓘ Infinitude of primes in the progression 1 mod 4 ⓘ Infinitude of primes in the progression 1 mod n for any n ⓘ Infinitude of primes in the progression 3 mod 4 ⓘ
statement	Every arithmetic progression a, a+d, a+2d, ... with gcd(a,d)=1 contains infinitely many prime numbers ⓘ
strengthenedBy	prime number theorem for arithmetic progressions ⓘ
typeOfResult	infinitude of primes result ⓘ
usesConcept	Dirichlet L-functions NERFINISHED ⓘ Dirichlet characters NERFINISHED ⓘ Euler product NERFINISHED ⓘ L-series NERFINISHED ⓘ characters modulo n ⓘ complex analysis ⓘ non-vanishing of L-functions at s = 1 ⓘ
yearProved	1837 ⓘ

Referenced by (7)

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

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet's theorem on arithmetic progressions ⓘ

Fermat's theorem on sums of two squares → relatedTo → Dirichlet's theorem on arithmetic progressions ⓘ

Multiplicative Number Theory → studiesProperty → Dirichlet's theorem on arithmetic progressions ⓘ

this entity surface form: Dirichlet’s theorem on primes in arithmetic progressions

Multiplicative Number Theory → hasClassicResult → Dirichlet's theorem on arithmetic progressions ⓘ

this entity surface form: Dirichlet’s theorem on arithmetic progressions

Chebotarev density theorem → implies → Dirichlet's theorem on arithmetic progressions ⓘ

Dirichlet L-functions → centralIn → Dirichlet's theorem on arithmetic progressions ⓘ

this entity surface form: Dirichlet’s theorem on arithmetic progressions

prime number theorem → relatedConcept → Dirichlet's theorem on arithmetic progressions ⓘ

this entity surface form: Dirichlet’s theorem on arithmetic progressions