# Fungrim entry: 312147

$\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi(k)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}$
TeX:
\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi(k)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Re$\operatorname{Re}(z)$ Real part
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("312147"),
Formula(LessEqual(Abs(Sub(DirichletL(s, chi), Sum(Div(chi(k), Pow(k, s)), For(k, 1, Sub(N, 1))))), HurwitzZeta(Re(s), N))),
Variables(q, chi, s, N),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1), Element(N, ZZGreaterEqual(1)))))

## Topics using this entry

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