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Fungrim entry: e4e707

θ3 ⁣(0,τ)=1+2n=1λ ⁣(n)qn1qn   where q=eπiτ\theta_{3}\!\left(0 , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{\lambda\!\left(n\right) {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:τH\tau \in \mathbb{H}
\theta_{3}\!\left(0 , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{\lambda\!\left(n\right) {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}

\tau \in \mathbb{H}
Fungrim symbol Notation Short description
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
LiouvilleLambdaλ ⁣(n)\lambda\!\left(n\right) Liouville function
Powab{a}^{b} Power
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
    Formula(Equal(JacobiTheta(3, 0, tau), Where(Add(1, Mul(2, Sum(Div(Mul(LiouvilleLambda(n), Pow(q, n)), Sub(1, Pow(q, n))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Assumptions(Element(tau, HH)))

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2019-09-22 15:43:45.410764 UTC