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Fungrim entry: 3ab92d

χGqχ(n)={φ(q),n1(modq)0,otherwise\sum_{\chi \in G_{q}} \chi(n) = \begin{cases} \varphi(q), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}
Assumptions:qZ1andnZq \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
Sumnf(n)\sum_{n} f(n) Sum
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Totientφ(n)\varphi(n) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZZ\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))))

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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