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

Dirichlet characters