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Fungrim entry: c1ffd4

G2k ⁣(τ)=ζ ⁣(2k)(m,n)Z2{(0,0)}gcd(m,n)=11(mτ+n)2kG_{2 k}\!\left(\tau\right) = \zeta\!\left(2 k\right) \sum_{\textstyle{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\} \atop \gcd\left(m, n\right) = 1}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}
Assumptions:kZ2andτHk \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
G_{2 k}\!\left(\tau\right) = \zeta\!\left(2 k\right) \sum_{\textstyle{\left(m, n\right) \in {\mathbb{Z}}^{2} \setminus \left\{\left(0, 0\right)\right\} \atop \gcd\left(m, n\right) = 1}} \frac{1}{{\left(m \tau + n\right)}^{2 k}}

k \in \mathbb{Z}_{\ge 2} \,\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(s) Riemann zeta function
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
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("c1ffd4"),
    Formula(Equal(EisensteinG(Mul(2, k), tau), Mul(RiemannZeta(Mul(2, k)), Sum(Div(1, Pow(Add(Mul(m, tau), n), Mul(2, k))), ForElement(Tuple(m, n), SetMinus(Pow(ZZ, 2), Set(Tuple(0, 0)))), Equal(GCD(m, n), 1))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH))))

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2019-10-05 13:11:19.856591 UTC