# Fungrim entry: 8f8fb7

$\left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1} \text{ with } \operatorname{Im}\!\left(\rho_{n}\right) < T\right) \;\implies\; \left(\lambda_{n} \ge 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 0} \text{ with } n \le {T}^{2}\right)$
Assumptions:$T \in \left[0, \infty\right)$
References:
• https://arxiv.org/abs/1703.02844
TeX:
\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)
Definitions:
Fungrim symbol Notation Short description
Re$\operatorname{Re}(z)$ Real part
RiemannZetaZero$\rho_{n}$ Nontrivial zero of the Riemann zeta function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Im$\operatorname{Im}(z)$ Imaginary part
KeiperLiLambda$\lambda_{n}$ Keiper-Li coefficient
Pow${a}^{b}$ Power
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("8f8fb7"),
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))))),
Variables(T),
Assumptions(Element(T, ClosedOpenInterval(0, Infinity))),
References("https://arxiv.org/abs/1703.02844"))

## Topics using this entry

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