DirichletCharacter

The $j$-th Dirichlet character modulo $k$ evaluated at $n$: a completely multiplicative, $k$-periodic map $\chi_j: \mathbb{Z} \to \mathbb{C}$, zero where $\gcd(n, k) > 1$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDirichlet character↗MathWorldDirichletCharacter↗WikidataQ1063579↗nLabDirichlet character↗DLMF27.8.p1↗FungrimDirichletCharacter✓↗Fungrim entry4cf4e4↗d8c6d1✓↗fc4f6a↗Wolfram LanguageDirichletCharacter!↗

DirichletCharacter(k, j, n)the character $\chi_j \bmod k$ at $n$, for $1 \le j \le \varphi(k)$; $j = 1$ is the principal character.

Typed signatures

Type	Package	Applies
(integer, integer, integer) -> number	compute-engine	always

Domain: Special functions

Details

• There are exactly $\varphi(k)$ characters mod $k$ — the dual group of $(\mathbb{Z}/k)^\times$ — and each is determined by its values on a set of generators. $\chi_1$, the principal character, is 1 on the units and 0 elsewhere.
• Every value is a root of unity of order dividing $\varphi(k)$, or 0: $\chi_j(mn) = \chi_j(m)\chi_j(n)$ always, and $\sum_{n=1}^{k}\chi_j(n) = 0$ for $j \ne 1$.
• The indexing is Wolfram's $\mathrm{DirichletCharacter}[k, j, n]$, which no published formula pins down; it is reproduced here by decomposing $(\mathbb{Z}/k)^\times$ into one cyclic factor per prime power of $k$ in ascending prime order (an odd $p^e$ contributing its least primitive root, $2^e$ contributing $\langle -1\rangle \times \langle 5 \rangle$) and reading $j - 1$ 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 $k$ with primitive root $g$ this comes out as $\chi_j(g) = e^{2\pi i (j-1)/(k-1)}$.
• An out-of-range $j$ (greater than $\varphi(k)$) names no character and is left unevaluated.
• A modulus past $2^{53}$ is left unevaluated too: the kernel works in machine integers.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

The nontrivial character mod 4: $\chi_2(3) = -1$ (this is the one behind DirichletBeta)

live ↗

In$\mathrm{DirichletCharacter}(4, 2, 3)$Out$-1$

$\chi_2(2) = i$ mod 5 — 2 is a primitive root, so it takes the generating fourth root of unity

live ↗

In$\mathrm{DirichletCharacter}(5, 2, 2)$Out$i$

Scope #

Values are roots of unity: 3 generates $(\mathbb{Z}/7)^\times$, and $\chi_3(3) = e^{2\pi i/3}$

live ↗

In$\mathrm{DirichletCharacter}(7, 3, 3)$Out$e^{\frac{2}{3}i π}$

Listable in $n$: the nontrivial character mod 3 over $1, \dots, 5$

live ↗

In$\mathrm{DirichletCharacter}(3, 2, \bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)$Out$\bigl\lbrack1, -1, 0, 1, -1\bigr\rbrack$

live ↗

In$\mathrm{DirichletCharacter}(20!, 300, 23)$Out$\mathrm{DirichletCharacter}(2\,432\,902\,008\,176\,640\,000, 300, 23)$

Properties #

A character vanishes where $\gcd(n, k) > 1$

live ↗

In$\mathrm{DirichletCharacter}(5, 2, 5)$Out$0$

$\chi_3$ mod 5 is the quadratic character: its period is $1, -1, -1, 1, 0$, the Legendre symbol $\left(\tfrac{n}{5}\right)$

live ↗

In$\mathrm{DirichletCharacter}(5, 3, 2)$Out$-1$

The trivial character mod 1 is 1 everywhere — its L-function is $\zeta$

live ↗

In$\mathrm{DirichletCharacter}(1, 1, 7)$Out$1$

Argument is reduced modulo the modulus

live ↗

In$\mathrm{DirichletCharacter}(7, 3, 20)$Out$1$

Product of two character values left unsimplified

live ↗

In$\mathrm{DirichletCharacter}(7, 3, 4)\mathrm{DirichletCharacter}(7, 3, 5)$Out$(-\frac{\sqrt{3}}{2}i-\frac{1}{2})(\frac{\sqrt{3}}{2}i-\frac{1}{2})$

Threads over a list, yielding a real character mod 12

live ↗

In$\mathrm{DirichletCharacter}(12, 3, \bigl\lbrack1, 5, 7, 11\bigr\rbrack)$Out$\bigl\lbrack1, 1, -1, -1\bigr\rbrack$

A value with an exact complex root of unity

live ↗

In$\mathrm{DirichletCharacter}(45, 8, 7)$Out$e^{\frac{5}{6}i π}$

A product of characters of different moduli, left unsimplified

live ↗

In$\mathrm{DirichletCharacter}(9, 2, 7)\mathrm{DirichletCharacter}(5, 4, 2)$Out$-i(-\frac{\sqrt{3}}{2}i-\frac{1}{2})$

Possible issues #

$j$ must be at most $\varphi(k) = 4$; out of range names no character and stays unevaluated

live ↗

In$\mathrm{DirichletCharacter}(5, 9, 2)$Out$\mathrm{DirichletCharacter}(5, 9, 2)$

About

Examples	15
Checked against	Wolfram Language 14 agree1 differ
Crosswalk	WikipediaMathWorldWikidatanLabDLMFFungrimFungrim entry ×3Wolfram Language

Implementation

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

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/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.

See also: DirichletL, DirichletBeta, DirichletEta, Zeta