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

Keiper-Li coefficients