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Fungrim entry: a4e947

n=0q1χ1(n)χ2(n)={φ(q),χ1=χ20,otherwise\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:qZ1  and  χ1Gq  and  χ2Gqq \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {\chi}_{1} \in G_{q} \;\mathbin{\operatorname{and}}\; {\chi}_{2} \in G_{q}
\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}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Conjugatez\overline{z} Complex conjugate
Totientφ(n)\varphi(n) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Source code for this entry:
    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)))))

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

2021-03-15 19:12:00.328586 UTC