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Fungrim entry: e2a734

(GRH)    (Re ⁣(ρn,χ)=12   for all (q,χ,n) with qZ1  and  χGq  and  nZ{0})\left(\operatorname{GRH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2} \;\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\}\right)
TeX:
\left(\operatorname{GRH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2} \;\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\}\right)
Definitions:
Fungrim symbol Notation Short description
GeneralizedRiemannHypothesisGRH\operatorname{GRH} Generalized Riemann hypothesis
ReRe(z)\operatorname{Re}(z) Real part
DirichletLZeroρn,χ\rho_{n,\chi} Nontrivial zero of Dirichlet L-function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("e2a734"),
    Formula(Equivalent(GeneralizedRiemannHypothesis, All(Equal(Re(DirichletLZero(n, chi)), Div(1, 2)), For(Tuple(q, chi, n)), And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, SetMinus(ZZ, Set(0))))))))

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2020-04-08 16:14:44.404316 UTC