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# Fungrim entry: 8ff1ff

$\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\right)$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1$
TeX:
\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\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
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
RiemannZeta$\zeta\!\left(s\right)$ Riemann 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("8ff1ff"),
Formula(LessEqual(Abs(DirichletL(s, chi)), RiemannZeta(Re(s)))),
Variables(q, chi, s),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1))))

## Topics using this entry

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

2020-04-08 16:14:44.404316 UTC