# Fungrim entry: e6deb7

$\left|\sum_{n=0}^{N} \chi\!\left(n\right)\right| \le \varphi\!\left(q\right)$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q}(1, \cdot)$
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
Abs$\left|z\right|$ Absolute value
Sum$\sum_{n} f\!\left(n\right)$ Sum
Totient$\varphi\!\left(n\right)$ Euler totient function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
ZZ$\mathbb{Z}$ Integers
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
DirichletCharacter$\chi_{q}(\ell, \cdot)$ 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)))))

## Topics using this entry

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