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Fungrim entry: 57d31a

χ ⁣(n){e2πik/r:kZ}{0}   where r=φ ⁣(q)\chi\!\left(n\right) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi\!\left(q\right)
Assumptions:qZ1andχGqandnZq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}
TeX:
\chi\!\left(n\right) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi\!\left(q\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
ZZZ\mathbb{Z} Integers
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("57d31a"),
    Formula(Where(Element(chi(n), Union(SetBuilder(Exp(Div(Mul(Mul(Mul(2, ConstPi), ConstI), k), r)), k, Element(k, ZZ)), Set(0))), Equal(r, Totient(q)))),
    Variables(q, chi, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, ZZ))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC