DirichletCharacter

The -th Dirichlet character modulo evaluated at : a completely multiplicative, -periodic map , zero where . Provided by `@enumeratio/analytic`.

DirichletCharacter(k, j, n)the character at , for ; is the principal character.

Domain: Special functions

Details

There are exactly characters mod — the dual group of — and each is determined by its values on a set of generators. , the principal character, is 1 on the units and 0 elsewhere.
Every value is a root of unity of order dividing , or 0: always, and for .
The indexing is Wolfram's , which no published formula pins down; it is reproduced here by decomposing into one cyclic factor per prime power of in ascending prime order (an odd contributing its least primitive root, contributing ) and reading as a mixed-radix exponent vector, first component most significant. The whole table is pinned against a Wolfram kernel for every modulus up to 40.
For a prime with primitive root this comes out as .
An out-of-range (greater than ) names no character and is left unevaluated.

Examples

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/analytic/src/dirichlet-l.ts

A discrete logarithm against the cyclic decomposition — an algorithm over mutable state, not a tree.

mappedexternalwolframpackages/oracle/src/mappings.ts

DirichletCharacter[k, j, n], same indexing. mpmath has no character indexing at all.