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Fungrim entry: e20db0

E6 ⁣(τ)=1504n=1σ5 ⁣(n)qn   where q=e2πiτE_{6}\!\left(\tau\right) = 1 - 504 \sum_{n=1}^{\infty} \sigma_{5}\!\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_{6}\!\left(\tau\right) = 1 - 504 \sum_{n=1}^{\infty} \sigma_{5}\!\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
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("e20db0"),
    Formula(Equal(EisensteinE(6, tau), Where(Sub(1, Mul(504, Sum(Mul(DivisorSigma(5, 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