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Fungrim entry: 9b3fde

L ⁣(s,χ)(q1+s2π)(1+ηRe(s))/2ζ ⁣(1+η)\left|L\!\left(s, \chi\right)\right| \le {\left(\frac{q \left|1 + s\right|}{2 \pi}\right)}^{\left( 1 + \eta - \operatorname{Re}\left(s\right) \right) / 2} \zeta\!\left(1 + \eta\right)
Assumptions:qZ2andχGqprimitiveandsCandη(0,12]andηRe ⁣(s)1+ηq \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \left(0, \frac{1}{2}\right] \,\mathbin{\operatorname{and}}\, -\eta \le \operatorname{Re}\!\left(s\right) \le 1 + \eta
References:
  • H. Rademacher, On the Phragmén-Lindelöf theorem and some applications, Mathematische Zeitschrift, December 1959, Volume 72, Issue 1, pp 192-204. Theorem 3. https://doi.org/10.1007/BF01162949
TeX:
\left|L\!\left(s, \chi\right)\right| \le {\left(\frac{q \left|1 + s\right|}{2 \pi}\right)}^{\left( 1 + \eta - \operatorname{Re}\left(s\right) \right) / 2} \zeta\!\left(1 + \eta\right)

q \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \eta \in \left(0, \frac{1}{2}\right] \,\mathbin{\operatorname{and}}\, -\eta \le \operatorname{Re}\!\left(s\right) \le 1 + \eta
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Source code for this entry:
Entry(ID("9b3fde"),
    Formula(LessEqual(Abs(DirichletL(s, chi)), Mul(Pow(Div(Mul(q, Abs(Add(1, s))), Mul(2, ConstPi)), Div(Sub(Add(1, eta), Re(s)), 2)), RiemannZeta(Add(1, eta))))),
    Variables(q, chi, s, eta),
    Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(chi, PrimitiveDirichletCharacters(q)), Element(s, CC), Element(eta, OpenClosedInterval(0, Div(1, 2))), LessEqual(Neg(eta), Re(s), Add(1, eta)))),
    References("H. Rademacher, On the Phragmén-Lindelöf theorem and some applications, Mathematische Zeitschrift, December 1959, Volume 72, Issue 1, pp 192-204. Theorem 3. https://doi.org/10.1007/BF01162949"))

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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