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Fungrim entry: 8ff1ff

L ⁣(s,χ)ζ ⁣(Re ⁣(s))\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right)\right)
Assumptions:qZ1andχGqandsCandRe ⁣(s)>1q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1
TeX:
\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right)\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}\!\left(s\right) > 1
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
CCC\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))))

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

2019-08-21 11:44:15.926409 UTC