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Fungrim entry: a50278

#zerosτFE2k ⁣(τ)1\# \mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{2 k}\!\left(\tau\right) \ge 1
Assumptions:kZ2k \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\# S Set cardinality
ZeroszerosP(x)f ⁣(x)\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right) Zeros (roots) of function
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
ZZGreaterEqualZn\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), tau, Element(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."))

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2019-08-17 11:32:46.829430 UTC