Fungrim entry: cae067

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2}\!\left(\tau\right) = \left\{ \tau + n : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{H},\,\operatorname{Re}(z) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(z\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2}\!\left(\tau\right) = \left\{ \tau + n : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{H},\,\operatorname{Re}(z) \in \left[-1 / 2, 1 / 2\right)} E_{2}\!\left(z\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \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
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("cae067"),
    Formula(Equal(Zeros(EisensteinE(2, tau), ForElement(tau, HH)), Set(Add(tau, n), For(Tuple(tau, n)), And(Element(tau, Zeros(EisensteinE(2, z), For(z), And(Element(z, HH), Element(Re(z), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2)))))), Element(n, ZZ))))))

Topics using this entry

Eisenstein series