Statements (102)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:item
|
| gptkbp:are_defined_on |
modular arithmetic
the integers |
| gptkbp:are_defined_over |
finite fields
|
| gptkbp:are_homomorphisms_from |
the multiplicative group of integers modulo n
|
| gptkbp:are_periodic_with_period |
the modulus
|
| gptkbp:are_used_to_prove |
the prime number theorem for arithmetic progressions
|
| gptkbp:can_be |
primitive or induced
|
| gptkbp:can_be_computed_using |
character tables
|
| gptkbp:can_be_evaluated_at |
integers
|
| gptkbp:can_be_extended_by |
characters of larger modulus
the whole group of integers |
| gptkbp:can_be_used_in |
the proof of the Chebotarev density theorem
|
| gptkbp:can_be_used_to_calculate |
L-functions at s=1
|
| gptkbp:can_be_used_to_derive |
congruences between L-values
results about the distribution of prime numbers. |
| gptkbp:can_be_used_to_prove |
the Chebotarev density theorem
the existence of infinitely many primes |
| gptkbp:constructed_in |
L-functions
characters of cyclic groups |
| gptkbp:generalize |
modular arithmetic
|
| gptkbp:has_applications_in |
gptkb:crypt
|
| gptkbp:has_role |
gptkb:Dirichlet's_theorem_on_primes_in_arithmetic_progressions
|
| https://www.w3.org/2000/01/rdf-schema#label |
Dirichlet characters
|
| gptkbp:is_analyzed_in |
the distribution of primes
the Riemann zeta function the arithmetic of elliptic curves the behavior of L-functions at critical points the behavior of L-functions at integers the behavior of primes in various settings the distribution of values of L-functions the symmetry of prime distributions the connections between number theory and representation theory |
| gptkbp:is_applied_in |
the study of elliptic curves
|
| gptkbp:is_associated_with |
Dirichlet L-functions
the distribution of quadratic residues |
| gptkbp:is_characterized_by |
their periodicity
their values on primitive roots |
| gptkbp:is_connected_to |
class field theory
the theory of modular forms Galois representations the theory of algebraic groups the Langlands program the theory of modular forms and elliptic curves the theory of motives |
| gptkbp:is_essential_for |
algebraic number theory
analytic number theory understanding the structure of the multiplicative group modulo n |
| gptkbp:is_evaluated_by |
the formula for L-functions
|
| gptkbp:is_explored_in |
the connections between number theory and geometry
the connections between number theory and algebraic topology |
| gptkbp:is_involved_in |
the study of modular forms
the study of Galois groups |
| gptkbp:is_related_to |
class field theory
Galois representations automorphic forms the Langlands program characters of finite groups the Riemann zeta function the theory of algebraic cycles the theory of cyclotomic fields the theory of p-adic forms the theory of representations of groups |
| gptkbp:is_represented_in |
Dirichlet series
characters of finite abelian groups characters of the group of units modulo n homomorphisms from the multiplicative group of integers modulo n |
| gptkbp:is_studied_for |
L-functions
the distribution of prime numbers the distribution of primes in arithmetic progressions the Galois group of number fields the arithmetic of abelian varieties the behavior of primes in arithmetic progressions |
| gptkbp:is_studied_in |
algebraic number theory
|
| gptkbp:is_used_in |
gptkb:crypt
gptkb:Dirichlet's_theorem_on_primes_in_arithmetic_progressions number theory the study of algebraic varieties modular forms analytic number theory the study of modular equations the study of modular forms the distribution of prime numbers the study of elliptic curves the study of rational points on curves the study of rational functions the study of automorphic forms the study of modular forms over finite fields the study of Galois cohomology the proof of the prime number theorem the study of cyclotomic fields the study of the arithmetic of function fields the study of the arithmetic of modular forms. |
| gptkbp:named_after |
gptkb:Peter_Gustav_Lejeune_Dirichlet
|
| gptkbp:related_to |
number theory
|
| gptkbp:scientific_classification |
their symmetry properties
imprimitive characters primitive characters their conductor their order |
| gptkbp:bfsParent |
gptkb:Peter_Gustav_Lejeune_Dirichlet
|
| gptkbp:bfsLayer |
7
|