Fungrim entry: e2a734

\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
`GeneralizedRiemannHypothesis`	$\operatorname{GRH}$	Generalized Riemann hypothesis
`Re`	$\operatorname{Re}(z)$	Real part
`DirichletLZero`	$\rho_{n,\chi}$	Nontrivial zero of Dirichlet L-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZ`	$\mathbb{Z}$	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))))))))

Fungrim entry: e2a734

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