Fungrim entry: a50278

\# \mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \ge 1

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) \ge 1

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

Definitions:

Fungrim symbol	Notation	Short description
`Cardinality`	$\# S$	Set cardinality
`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
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a50278"),
    Formula(GreaterEqual(Cardinality(Zeros(EisensteinE(Mul(2, k), tau), ForElement(tau, ModularGroupFundamentalDomain))), 1)),
    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