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Fungrim entry: 4877d1

n=0q1χ(n)={φ(q),χ=χq.10,otherwise\sum_{n=0}^{q - 1} \chi(n) = \begin{cases} \varphi(q), & \chi = \chi_{q \, . \, 1}\\0, & \text{otherwise}\\ \end{cases}
Assumptions:qZ1andχGqq \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
Sumnf(n)\sum_{n} f(n) Sum
Totientφ(n)\varphi(n) Euler totient function
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{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)))))

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

2019-10-05 13:11:19.856591 UTC