``mathxlab.nt.dirichlet``
=========================

Dirichlet characters and small numerical helpers.

.. admonition:: Stability
   :class: note

   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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: python

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

Public API
----------

.. list-table::
   :header-rows: 1
   :widths: 22 10 68

   * - 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
~~~~~~~

.. autoclass:: mathxlab.nt.dirichlet.DirichletCharacter
   :members:
   :member-order: bysource
   :show-inheritance:

Functions
~~~~~~~~~

.. autofunction:: mathxlab.nt.dirichlet.euler_phi

.. autofunction:: mathxlab.nt.dirichlet.reduced_residues

.. autofunction:: mathxlab.nt.dirichlet.all_characters

.. autofunction:: mathxlab.nt.dirichlet.character_table

.. autofunction:: mathxlab.nt.dirichlet.conductor

.. autofunction:: mathxlab.nt.dirichlet.orthogonality_matrix