Fungrim entry: 4877d1

\sum_{n=0}^{q - 1} \chi(n) = \begin{cases} \varphi(q), & \chi = \chi_{q \, . \, 1}\\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(n) = \begin{cases} \varphi(q), & \chi = \chi_{q \, . \, 1}\\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`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	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), For(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

Dirichlet characters