Fungrim home page

Fungrim entry: cae067

zerosτHE2 ⁣(τ)={τ+n:τzeroszH,Re(z)[1/2,1/2)E2 ⁣(z)  and  nZ}\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\}
\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\}
Fungrim symbol Notation Short description
ZeroszerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) 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}(z) Real part
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
ZZZ\mathbb{Z} Integers
Source code for this entry:
    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

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

2021-03-15 19:12:00.328586 UTC