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Fungrim entry: e6deb7

n=0Nχ(n)φ(q)\left|\sum_{n=0}^{N} \chi(n)\right| \le \varphi(q)
Assumptions:qZ1  and  NZ  and  χGq  and  χχq.1q \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}
\left|\sum_{n=0}^{N} \chi(n)\right| \le \varphi(q)

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}
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Sumnf(n)\sum_{n} f(n) Sum
Totientφ(n)\varphi(n) 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} Dirichlet character
Source code for this entry:
    Formula(LessEqual(Abs(Sum(chi(n), For(n, 0, N))), Totient(q))),
    Variables(N, chi, q),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(N, ZZ), Element(chi, DirichletGroup(q)), NotEqual(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.

2021-03-15 19:12:00.328586 UTC