Fungrim entry: 3ab92d

\sum_{\chi \in G_{q}} \chi(n) = \begin{cases} \varphi(q), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\sum_{\chi \in G_{q}} \chi(n) = \begin{cases} \varphi(q), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("3ab92d"),
    Formula(Equal(Sum(chi(n), ForElement(chi, DirichletGroup(q))), Cases(Tuple(Totient(q), CongruentMod(n, 1, q)), Tuple(0, Otherwise)))),
    Variables(q, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(n, ZZ))))

Topics using this entry

Dirichlet characters