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Fungrim entry: 62f12c

Gq ⁣(χ)=n=1qχ(n)e2πin/qG_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi(n) {e}^{2 \pi i n / q}
Assumptions:qZ1andχGqq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
TeX:
G_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi(n) {e}^{2 \pi i n / q}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
Fungrim symbol Notation Short description
GaussSumGq ⁣(χ)G_{q}\!\left(\chi\right) Gauss sum
Sumnf(n)\sum_{n} f(n) Sum
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
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("62f12c"),
    Formula(Equal(GaussSum(q, chi), Sum(Mul(chi(n), Exp(Div(Mul(Mul(Mul(2, Pi), ConstI), n), q))), For(n, 1, q)))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

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

2019-12-11 23:01:54.699850 UTC