# Fungrim entry: f4de66

$\sum_{\chi \in G_{q}} \chi(m) \overline{\chi(n)} = \begin{cases} \varphi(q), & n \equiv m \pmod {q} \;\mathbin{\operatorname{and}}\; \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}$
TeX:
\sum_{\chi \in G_{q}} \chi(m) \overline{\chi(n)} = \begin{cases} \varphi(q), & n \equiv m \pmod {q} \;\mathbin{\operatorname{and}}\; \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Conjugate$\overline{z}$ Complex conjugate
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
Totient$\varphi(n)$ Euler totient function
GCD$\gcd\!\left(a, b\right)$ Greatest common divisor
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("f4de66"),
Formula(Equal(Sum(Mul(chi(m), Conjugate(chi(n))), ForElement(chi, DirichletGroup(q))), Cases(Tuple(Totient(q), And(CongruentMod(n, m, q), Equal(GCD(m, q), 1))), Tuple(0, Otherwise)))),
Variables(q, m, n),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(m, ZZ), Element(n, ZZ))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC