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# Fungrim entry: e4e707

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

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Sum$\sum_{n} f(n)$ Sum
LiouvilleLambda$\lambda(n)$ Liouville function
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("e4e707"),
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(Pi, ConstI), tau)))))),
Variables(tau),
Assumptions(Element(tau, HH)))

## Topics using this entry

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

2020-01-31 18:09:28.494564 UTC