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The most basic theta series is defined by $\theta(z)=\sum_{n=-\infty}^{\infty}q^{n^2}$, where $q=e^{2\pi iz}$. This is connected to the question of how many representatations does an integer $n$ have as a sum of $k$ squares. The answer is just the $n$-th coefficient of $\theta(z)^k$.
This coefficent can be (at least in some cases) computed thanks to an astonsihing fact -- $\theta(z)$ is a modular form of weight $1/2$. The can be obtained by Fourier-transforming the function $f(t)=e^{-at^2}$. The result is $\frac 1 {2\sqrt a}e^{-\omega^2/4a}$, also a Gausian function.

But what if we try to analyze sums of higher powers, 3rd or even 4th. The function $\sum_{n=-\infty}^{\infty}q^{n^3}$ is not holomorphic, but $\sum_{n=-\infty}^{\infty}q^{n^4}$ is. However, for the Fourier transform of $f(t)=e^{-at^4}$ we get the DE: $\frac {d^3F} {d\omega^3}=\frac {\omega} {4a}F$. This is much more complicated than the original case and the function is certainly not a modular form.

But since the only difference between the two functions is the power of $n$, this naturally suggests that there may be at least some nice behaviour under some transformation. The function is certainly periodic with period 1, but that doesn't really help. Is there a more general framework, in which these functions may be studied and questions like "how many times can an integer be represented as $a^4+b^4$" answered?

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    $\begingroup$ See also this question: mathoverflow.net/questions/57717/… $\endgroup$ – Stopple Mar 4 at 22:27
  • $\begingroup$ $\sum_n e^{2i \pi n^4 z} = \sum_{m \ge 1} \lambda(m) \theta(m^2 z)$ where $ \lambda(m) = \sum_{d^2 | m} \mu(m/d^2)$ and $\sum_n e^{-2i \pi n^4 / z} = (4iz)^{1/2} \sum_{m \ge 1} \lambda(m) m^{-1} \theta(z/ m^2)$ which isn't periodic. $\endgroup$ – reuns Mar 5 at 5:37

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