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Fungrim entry: f33f09

zerosτH,Re(τ)[1/2,1/2)E2 ⁣(τ)={(zero*zD(c,d)E2 ⁣(z)):cZanddZandgcd ⁣(c,d)=1anddc[12,12)}   where D ⁣(c,d)=ClosedDisk ⁣(dc+iπ6c2,0.000283π236c2)\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H},\,\operatorname{Re}\left(\tau\right) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(\tau\right) = \left\{ \left(\mathop{\operatorname{zero*}\,}\limits_{z \in D\left(c, d\right)} E_{2}\!\left(z\right)\right) : c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \gcd\!\left(c, d\right) = 1 \,\mathbin{\operatorname{and}}\, -\frac{d}{c} \in \left[-\frac{1}{2}, \frac{1}{2}\right) \right\}\; \text{ where } D\!\left(c, d\right) = \operatorname{ClosedDisk}\!\left(-\frac{d}{c} + \frac{i \pi}{6 {c}^{2}}, \frac{0.000283 {\pi}^{2}}{36 {c}^{2}}\right)
References:
  • R. Wood and M. P. Young, Zeros of the weight two Eisenstein series, Journal of Number Theory Volume 143, October 2014, Pages 320-333. https://doi.org/10.1016/j.jnt.2014.04.007
TeX:
\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H},\,\operatorname{Re}\left(\tau\right) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(\tau\right) = \left\{ \left(\mathop{\operatorname{zero*}\,}\limits_{z \in D\left(c, d\right)} E_{2}\!\left(z\right)\right) : c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \gcd\!\left(c, d\right) = 1 \,\mathbin{\operatorname{and}}\, -\frac{d}{c} \in \left[-\frac{1}{2}, \frac{1}{2}\right) \right\}\; \text{ where } D\!\left(c, d\right) = \operatorname{ClosedDisk}\!\left(-\frac{d}{c} + \frac{i \pi}{6 {c}^{2}}, \frac{0.000283 {\pi}^{2}}{36 {c}^{2}}\right)
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
HHH\mathbb{H} Upper complex half-plane
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
UniqueZerozero*P(x)f ⁣(x)\mathop{\operatorname{zero*}\,}\limits_{P\left(x\right)} f\!\left(x\right) Unique zero (root) of function
ZZZ\mathbb{Z} Integers
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ConstIii Imaginary unit
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
Source code for this entry:
Entry(ID("f33f09"),
    Formula(Equal(Zeros(EisensteinE(2, tau), Var(tau), And(Element(tau, HH), Element(Re(tau), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2))))), Where(SetBuilder(Parentheses(UniqueZero(EisensteinE(2, z), Var(z), Element(z, D(c, d)))), Tuple(c, d), And(Element(c, ZZ), Element(d, ZZ), Equal(GCD(c, d), 1), Element(Neg(Div(d, c)), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2))))), Equal(D(c, d), ClosedDisk(Add(Neg(Div(d, c)), Div(Mul(ConstI, ConstPi), Mul(6, Pow(c, 2)))), Div(Mul(Decimal("0.000283"), Pow(ConstPi, 2)), Mul(36, Pow(c, 2)))))))),
    References("R. Wood and M. P. Young, Zeros of the weight two Eisenstein series, Journal of Number Theory Volume 143, October 2014, Pages 320-333. https://doi.org/10.1016/j.jnt.2014.04.007"))

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

2019-09-22 15:43:45.410764 UTC