# Fungrim entry: 15b347

$E_{2 k}\!\left(\tau\right) = 1 - \frac{4 k}{B_{2 k}} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} {n}^{2 k - 1} {q}^{m n}\; \text{ where } q = {e}^{2 \pi i \tau}$
Assumptions:$k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H}$
TeX:
E_{2 k}\!\left(\tau\right) = 1 - \frac{4 k}{B_{2 k}} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} {n}^{2 k - 1} {q}^{m 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
BernoulliB$B_{n}$ Bernoulli number
Sum$\sum_{n} f(n)$ Sum
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("15b347"),
Formula(Equal(EisensteinE(Mul(2, k), tau), Where(Sub(1, Mul(Div(Mul(4, k), BernoulliB(Mul(2, k))), Sum(Sum(Mul(Pow(n, Sub(Mul(2, k), 1)), Pow(q, Mul(m, n))), For(m, 1, Infinity)), 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.

2021-03-15 19:12:00.328586 UTC