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Fungrim entry: e6deb7

n=0Nχ ⁣(n)φ ⁣(q)\left|\sum_{n=0}^{N} \chi\!\left(n\right)\right| \le \varphi\!\left(q\right)
Assumptions:qZ1andNZandχGqandχχq(1,)q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q}(1, \cdot)
TeX:
\left|\sum_{n=0}^{N} \chi\!\left(n\right)\right| \le \varphi\!\left(q\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q}(1, \cdot)
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZZ\mathbb{Z} Integers
DirichletGroupGqG_{q} Dirichlet characters with given modulus
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
Source code for this entry:
Entry(ID("e6deb7"),
    Formula(LessEqual(Abs(Sum(chi(n), Tuple(n, 0, N))), Totient(q))),
    Variables(N),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(N, ZZ), Element(chi, DirichletGroup(q)), Unequal(chi, DirichletCharacter(q, 1)))))

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

2019-08-19 14:38:23.809000 UTC