polythetarithms

\theta_{3}(z,q)=\sum_{n\in\mathbb{N}}q^{n^{2}}e^{2niz}

\theta_{3}(z,q)=\sum_{n\in\mathbb{N}}q^{n^{2}}\sum_{m=1}^{\infty}\frac{(2niz)^{m}}{m!}=\sum_{m=1}^{\infty}\frac{(2iz)^{m}}{m!}\sum_{n\in\mathbb{N}}\frac{q^{n^{2}}}{n^{-m}}

I asked about something like this a while ago on math.stackexchange: 

http://math.stackexchange.com/questions/96868/regularity-of-root-spacing-of-gz-sum-limits-n-1-infty-frace-n2

for the moment, let’s call:

\Lambda\mathrm{i}_{s}(q)\equiv\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{n^{s}} the polythetarithm

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