Fungrim entry: f33f09

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H},\,\operatorname{Re}(\tau) \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}(\tau) \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
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane
`Re`	$\operatorname{Re}(z)$	Real part
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("f33f09"),
    Formula(Equal(Zeros(EisensteinE(2, tau), ForElement(tau, HH), Element(Re(tau), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2)))), Where(Set(Parentheses(UniqueZero(EisensteinE(2, z), ForElement(z, D(c, d)))), For(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, Pi), Mul(6, Pow(c, 2)))), Div(Mul(Decimal("0.000283"), Pow(Pi, 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"))

Topics using this entry

Eisenstein series