Fungrim entry: 214a91

\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \;\mathbin{\operatorname{and}}\; \left(\left(q < 400000 \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}\!\left(\rho_{n,\chi}\right)\right| < \frac{{10}^{8}}{q}\right) \;\mathbin{\operatorname{or}}\; \operatorname{GRH}\right)

References:

D. J. Platt (2013), Numerical computations concerning the GRH. https://arxiv.org/pdf/1305.3087.pdf

TeX:

\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \;\mathbin{\operatorname{and}}\; \left(\left(q < 400000 \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}\!\left(\rho_{n,\chi}\right)\right| < \frac{{10}^{8}}{q}\right) \;\mathbin{\operatorname{or}}\; \operatorname{GRH}\right)

Definitions:

Fungrim symbol	Notation	Short description
`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
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`ZZ`	$\mathbb{Z}$	Integers
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Pow`	${a}^{b}$	Power
`GeneralizedRiemannHypothesis`	$\operatorname{GRH}$	Generalized Riemann hypothesis

Source code for this entry:

Entry(ID("214a91"),
    Formula(Equal(Re(DirichletLZero(n, chi)), Div(1, 2))),
    Variables(q, n, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, PrimitiveDirichletCharacters(q)), Element(n, ZZ), NotEqual(n, 0), Or(And(Less(q, 400000), Less(Abs(Im(DirichletLZero(n, chi))), Div(Pow(10, 8), q))), GeneralizedRiemannHypothesis))),
    References("D. J. Platt (2013), Numerical computations concerning the GRH. https://arxiv.org/pdf/1305.3087.pdf"))

Topics using this entry

Dirichlet L-functions