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Fungrim entry: 2f6805

zerosτFE2k ⁣(τ){eiθ:θ[π2,2π3]}\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: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) \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
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
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
ClosedInterval[a,b]\left[a, b\right] Closed interval
ConstPiπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\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), tau, Element(tau, ModularGroupFundamentalDomain)), SetBuilder(Exp(Mul(ConstI, theta)), theta, Element(theta, ClosedInterval(Div(ConstPi, 2), Div(Mul(2, ConstPi), 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."))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC