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Fungrim entry: 7c00e6

E2k ⁣(τ)=14nB2nn=1σ2k1 ⁣(n)qn   where q=e2πiτE_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \sum_{n=1}^{\infty} \sigma_{2 k - 1}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{2 \pi i \tau}
Assumptions:kZ1andτHk \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
E_{2 k}\!\left(\tau\right) = 1 - \frac{4 n}{B_{2 n}} \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
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
BernoulliBBnB_{n} Bernoulli number
Sumnf(n)\sum_{n} f(n) Sum
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
Powab{a}^{b} Power
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
HHH\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, n), BernoulliB(Mul(2, n))), Sum(Mul(DivisorSigma(Sub(Mul(2, k), 1), n), Pow(q, n)), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Mul(2, ConstPi), ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

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2019-10-05 13:11:19.856591 UTC