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Fungrim entry: 8f8fb7

(Re ⁣(ρn)=12   for all nZ1 with Im ⁣(ρn)<T)        (λn0   for all nZ0 with nT2)\left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1} \text{ with } \operatorname{Im}\!\left(\rho_{n}\right) < T\right) \;\implies\; \left(\lambda_{n} \ge 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 0} \text{ with } n \le {T}^{2}\right)
Assumptions:T[0,)T \in \left[0, \infty\right)
\left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1} \text{ with } \operatorname{Im}\!\left(\rho_{n}\right) < T\right) \;\implies\; \left(\lambda_{n} \ge 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 0} \text{ with } n \le {T}^{2}\right)

T \in \left[0, \infty\right)
Fungrim symbol Notation Short description
ReRe(z)\operatorname{Re}(z) Real part
RiemannZetaZeroρn\rho_{n} Nontrivial zero of the Riemann zeta function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ImIm(z)\operatorname{Im}(z) Imaginary part
KeiperLiLambdaλn\lambda_{n} Keiper-Li coefficient
Powab{a}^{b} Power
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Implies(All(Equal(Re(RiemannZetaZero(n)), Div(1, 2)), ForElement(n, ZZGreaterEqual(1)), Less(Im(RiemannZetaZero(n)), T)), All(GreaterEqual(KeiperLiLambda(n), 0), ForElement(n, ZZGreaterEqual(0)), LessEqual(n, Pow(T, 2))))),
    Assumptions(Element(T, ClosedOpenInterval(0, Infinity))),

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

2021-03-15 19:12:00.328586 UTC