mathxlab.nt.dirichlet

Dirichlet characters and small numerical helpers.

Stability

Status: Experimental.

This project treats the documented names as the public surface, but details may still evolve. If you need strict API guarantees, add __all__ = [...] to each module and version releases accordingly.

Design notes

• Functions are designed for experiment-scale inputs (not cryptographic workloads).

• Prefer explicit parameters (e.g., n_max) for reproducibility.

Examples

Evaluate a Dirichlet character

from mathxlab.nt.dirichlet import all_characters
chi = all_characters(5)[1]
print(chi(2))

Public API

Name	Kind	Summary
euler_phi	function	Compute Euler’s totient φ(n) by prime factorization.
reduced_residues	function	Return the reduced residue system modulo q (sorted).
DirichletCharacter	class	Dirichlet character modulo q.
all_characters	function	Enumerate all Dirichlet characters modulo q.
character_table	function	Return the full character table as a matrix.
conductor	function	Compute the conductor of a character by brute-force divisor checks.
orthogonality_matrix	function	Compute the character orthogonality matrix for modulus q.

Reference

Classes

class mathxlab.nt.dirichlet.DirichletCharacter(modulus, params, _components)

Bases: object

Dirichlet character modulo q.

The character is stored as a product of prime-power component characters.

modulus

Modulus q.

params

Parameter tuples, one per prime-power component.

Examples

>>> from mathxlab.nt.dirichlet import DirichletCharacter
>>> DirichletCharacter

property is_principal

Return True if χ is the principal character.

table()

Return the character table row for residues 0..q-1.

Returns:

A complex NumPy array of shape (q,) containing χ(0), χ(1), …, χ(q-1).

property conductor

Return the conductor of the character.

Returns:

The smallest f | q such that χ(n) depends only on n mod f.

property is_primitive

Return True if the character is primitive.

Returns:

True iff conductor(χ) == q.

Functions

mathxlab.nt.dirichlet.euler_phi(n)

Compute Euler’s totient φ(n) by prime factorization.

Parameters:

n – Positive integer.

Returns:

φ(n).

Examples

>>> from mathxlab.nt.dirichlet import euler_phi
>>> euler_phi

mathxlab.nt.dirichlet.reduced_residues(q)

Return the reduced residue system modulo q (sorted).

Parameters:

q – Modulus (>= 1).

Returns:

List of a in [1, q] with gcd(a, q) = 1.

Examples

>>> from mathxlab.nt.dirichlet import reduced_residues
>>> reduced_residues

mathxlab.nt.dirichlet.all_characters(q)

Enumerate all Dirichlet characters modulo q.

Parameters:

q – Modulus (>= 1).

Returns:

List of DirichletCharacter objects of length φ(q).

Examples

>>> from mathxlab.nt.dirichlet import all_characters
>>> chars = all_characters(5)
>>> len(chars)
4

mathxlab.nt.dirichlet.character_table(q)

Return the full character table as a matrix.

Parameters:

q – Modulus.

Returns:

Complex matrix of shape (φ(q), q) where rows are characters and columns are residues 0..q-1.

Examples

>>> from mathxlab.nt.dirichlet import character_table
>>> character_table

mathxlab.nt.dirichlet.conductor(chi)

Compute the conductor of a character by brute-force divisor checks.

The conductor is the smallest f | q such that χ(n) depends only on n mod f.

This routine is intended for small q and experiment use.

Parameters:

chi – Character modulo q.

Returns:

The conductor f.

Examples

>>> from mathxlab.nt.dirichlet import conductor
>>> conductor

mathxlab.nt.dirichlet.orthogonality_matrix(q)

Compute the character orthogonality matrix for modulus q.

The matrix entries are:

M[i,j] = (1/φ(q)) * ∑_{a mod q, gcd(a,q)=1} χ_i(a) conj(χ_j(a))

For a correct implementation, M should be the identity.

Parameters:

q – Modulus.

Returns:

Complex matrix of shape (φ(q), φ(q)).

Examples

>>> from mathxlab.nt.dirichlet import orthogonality_matrix
>>> orthogonality_matrix