# Fungrim entry: 4877d1

$\sum_{n=0}^{q - 1} \chi\!\left(n\right) = \begin{cases} \varphi\!\left(q\right), & \chi = \chi_{q}(1, \cdot)\\0, & \text{otherwise}\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}$
TeX:
\sum_{n=0}^{q - 1} \chi\!\left(n\right) = \begin{cases} \varphi\!\left(q\right), & \chi = \chi_{q}(1, \cdot)\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f\!\left(n\right)$ Sum
Totient$\varphi\!\left(n\right)$ Euler totient function
DirichletCharacter$\chi_{q}(\ell, \cdot)$ Dirichlet character
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("4877d1"),
Formula(Equal(Sum(chi(n), Tuple(n, 0, Sub(q, 1))), Cases(Tuple(Totient(q), Equal(chi, DirichletCharacter(q, 1))), Tuple(0, Otherwise)))),
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.

2019-08-19 14:38:23.809000 UTC