Fungrim entry: f8dfaf

$E_{4}\!\left(\tau\right) = 1 + 240 \sum_{n=1}^{\infty} \sigma_{3}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}$
Assumptions:$k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
E_{4}\!\left(\tau\right) = 1 + 240 \sum_{n=1}^{\infty} \sigma_{3}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
EisensteinE$E_{k}\!\left(\tau\right)$ Normalized Eisenstein series
Sum$\sum_{n} f(n)$ Sum
DivisorSigma$\sigma_{k}\!\left(n\right)$ Sum of divisors function
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("f8dfaf"),
Formula(Equal(EisensteinE(4, tau), Where(Add(1, Mul(240, Sum(Mul(DivisorSigma(3, n), Pow(q, n)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, Pi), ConstI), tau)))))),
Variables(k, tau),
Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

Topics using this entry

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

2020-08-27 09:56:25.682319 UTC