Fungrim entry: 2f6805

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \subset \left\{ {e}^{i \theta} : \theta \in \left[\frac{\pi}{2}, \frac{2 \pi}{3}\right] \right\}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

References:

F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein Series, Bull. London Math. Soc., 2(1970),169-170.

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \subset \left\{ {e}^{i \theta} : \theta \in \left[\frac{\pi}{2}, \frac{2 \pi}{3}\right] \right\}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("2f6805"),
    Formula(Subset(Zeros(EisensteinE(Mul(2, k), tau), ForElement(tau, ModularGroupFundamentalDomain)), Set(Exp(Mul(ConstI, theta)), ForElement(theta, ClosedInterval(Div(Pi, 2), Div(Mul(2, Pi), 3)))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)))),
    References("F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein Series, Bull. London Math. Soc., 2(1970),169-170."))

Topics using this entry

Eisenstein series