Fungrim home page

Fungrim entry: e2a734

(GRH)    (for all (q,χ,n) with qZ1andχGqandnZ{0},Re ⁣(ρn,χ)=12)\left(\operatorname{GRH}\right) \iff \left(\text{for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \setminus \left\{0\right\}, \operatorname{Re}\!\left(\rho_{n, \chi}\right) = \frac{1}{2}\right)
TeX:
\left(\operatorname{GRH}\right) \iff \left(\text{for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \setminus \left\{0\right\}, \operatorname{Re}\!\left(\rho_{n, \chi}\right) = \frac{1}{2}\right)
Definitions:
Fungrim symbol Notation Short description
GeneralizedRiemannHypothesisGRH\operatorname{GRH} Generalized Riemann hypothesis
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
ZZZ\mathbb{Z} Integers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
DirichletLZeroρn,χ\rho_{n, \chi} Nontrivial zero of Dirichlet L-function
Source code for this entry:
Entry(ID("e2a734"),
    Formula(Equivalent(GeneralizedRiemannHypothesis, ForAll(Tuple(q, chi, n), And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, SetMinus(ZZ, Set(0)))), Equal(Re(DirichletLZero(n, chi)), Div(1, 2))))))

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