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Fungrim entry: 0a2120

E2k ⁣(τ)=G2k ⁣(τ)2ζ ⁣(2k)E_{2 k}\!\left(\tau\right) = \frac{G_{2 k}\!\left(\tau\right)}{2 \zeta\!\left(2 k\right)}
Assumptions:kZ1andτHk \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
E_{2 k}\!\left(\tau\right) = \frac{G_{2 k}\!\left(\tau\right)}{2 \zeta\!\left(2 k\right)}

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
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
RiemannZetaζ(s)\zeta(s) Riemann zeta function
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("0a2120"),
    Formula(Equal(EisensteinE(Mul(2, k), tau), Div(EisensteinG(Mul(2, k), tau), Mul(2, RiemannZeta(Mul(2, k)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

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

2019-10-05 13:11:19.856591 UTC