Fungrim entry: 7c00e6

E_{2 k}\!\left(\tau\right) = 1 - \frac{4 k}{B_{2 k}} \sum_{n=1}^{\infty} \sigma_{2 k - 1}\!\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_{2 k}\!\left(\tau\right) = 1 - \frac{4 k}{B_{2 k}} \sum_{n=1}^{\infty} \sigma_{2 k - 1}\!\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
`BernoulliB`	$B_{n}$	Bernoulli number
`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("7c00e6"),
    Formula(Equal(EisensteinE(Mul(2, k), tau), Where(Sub(1, Mul(Div(Mul(4, k), BernoulliB(Mul(2, k))), Sum(Mul(DivisorSigma(Sub(Mul(2, k), 1), 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

Eisenstein series