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Fungrim entry: 214a91

Re ⁣(ρn,χ)=12\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2}
Assumptions:qZ1  and  χGqprimitive  and  nZ  and  n0  and  ((q<400000  and  Im ⁣(ρn,χ)<108q)  or  GRH)q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}^{\text{primitive}} \;\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_{q}^{\text{primitive}} \;\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
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
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
ZZZ\mathbb{Z} Integers
Absz\left|z\right| Absolute value
ImIm(z)\operatorname{Im}(z) Imaginary part
Powab{a}^{b} Power
GeneralizedRiemannHypothesisGRH\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"))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC