Fungrim entry: 62f12c

G_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi(n) {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(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
`GaussSum`	$G_{q}\!\left(\chi\right)$	Gauss sum
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	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, Pi), ConstI), n), q))), For(n, 1, q)))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

Topics using this entry

Dirichlet L-functions