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Fungrim entry: b07750

G2k ⁣(τ)=2ζ ⁣(2k)+2m=1nZ1(mτ+n)2kG_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right) + 2 \sum_{m=1}^{\infty} \sum_{n \in \mathbb{Z}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}
Assumptions:kZ1andτHk \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
G_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right) + 2 \sum_{m=1}^{\infty} \sum_{n \in \mathbb{Z}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}

k \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
Infinity\infty Positive infinity
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("b07750"),
    Formula(Equal(EisensteinG(Mul(2, k), tau), Add(Mul(2, RiemannZeta(Mul(2, k))), Mul(2, Sum(Sum(Div(1, Pow(Add(Mul(m, tau), n), Mul(2, k))), n, Element(n, ZZ)), Tuple(m, 1, Infinity)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(tau, HH))))

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2019-08-17 11:32:46.829430 UTC