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

Dirichlet characters