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Fungrim entry: f4de66

χGqχ(m)χ(n)={φ(q),nm(modq)  and  gcd ⁣(m,q)=10,otherwise\sum_{\chi \in G_{q}} \chi(m) \overline{\chi(n)} = \begin{cases} \varphi(q), & n \equiv m \pmod {q} \;\mathbin{\operatorname{and}}\; \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}
Assumptions:qZ1  and  mZ  and  nZq \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
TeX:
\sum_{\chi \in G_{q}} \chi(m) \overline{\chi(n)} = \begin{cases} \varphi(q), & n \equiv m \pmod {q} \;\mathbin{\operatorname{and}}\; \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Conjugatez\overline{z} Complex conjugate
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Totientφ(n)\varphi(n) Euler totient function
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("f4de66"),
    Formula(Equal(Sum(Mul(chi(m), Conjugate(chi(n))), ForElement(chi, DirichletGroup(q))), Cases(Tuple(Totient(q), And(CongruentMod(n, m, q), Equal(GCD(m, q), 1))), Tuple(0, Otherwise)))),
    Variables(q, m, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(m, ZZ), 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.

2020-08-27 09:56:25.682319 UTC