# Fungrim entry: a4e947

$\sum_{n=0}^{q - 1} {\chi}_{1}(n) \overline{{\chi}_{2}(n)} = \begin{cases} \varphi(q), & {\chi}_{1} = {\chi}_{2}\\0, & \text{otherwise}\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {\chi}_{1} \in G_{q} \;\mathbin{\operatorname{and}}\; {\chi}_{2} \in G_{q}$
TeX:
\sum_{n=0}^{q - 1} {\chi}_{1}(n) \overline{{\chi}_{2}(n)} = \begin{cases} \varphi(q), & {\chi}_{1} = {\chi}_{2}\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {\chi}_{1} \in G_{q} \;\mathbin{\operatorname{and}}\; {\chi}_{2} \in G_{q}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Conjugate$\overline{z}$ Complex conjugate
Totient$\varphi(n)$ Euler totient function
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("a4e947"),
Formula(Equal(Sum(Mul(Subscript(chi, 1)(n), Conjugate(Subscript(chi, 2)(n))), For(n, 0, Sub(q, 1))), Cases(Tuple(Totient(q), Equal(Subscript(chi, 1), Subscript(chi, 2))), Tuple(0, Otherwise)))),
Variables(q, Subscript(chi, 1), Subscript(chi, 2)),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(Subscript(chi, 1), DirichletGroup(q)), Element(Subscript(chi, 2), 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.

2020-08-27 09:56:25.682319 UTC