# Fungrim entry: 62f12c

$G_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi\!\left(n\right) {e}^{2 \pi i n / q}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}$
TeX:
G_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi\!\left(n\right) {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
GaussSum$G_{q}\!\left(\chi\right)$ Gauss sum
Sum$\sum_{n} f\!\left(n\right)$ Sum
Exp${e}^{z}$ Exponential function
ConstPi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ 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, ConstPi), ConstI), n), q))), Tuple(n, 1, q)))),
Variables(q, chi),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

## Topics using this entry

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