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I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at least an upperbound of the type $|f(k)|\leq \frac{C}{k^p}$ for some constant $C$ and power $p$?

Thanks in advance!

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    $\begingroup$ The upper bound is pretty easy: $f(k) \leq \int_0^\infty e^{-kx^2} \, dx = \frac{1}{2} \sqrt{\pi/k}$ since $f(k)$ is concave up. I doubt there's a simple closed form expression in elementary functions, but there could be using special functions. $\endgroup$ – Jair Taylor May 21 '15 at 19:45
  • $\begingroup$ Aha of course, I overlooked that possibility. I am actually interested in a better bound than $1/\sqrt{k}$. Something like $1/k^{1/2+\varepsilon}$ with $\varepsilon \in (0,1/2)$. But I don't know if it trivially seen. $\endgroup$ – Martingalo May 21 '15 at 19:55
  • $\begingroup$ Do you want the estimate for all $k>0$ or for, say, $k\ge 1$? $\endgroup$ – zhw. May 21 '15 at 19:58
  • $\begingroup$ if it works for $k\geq 1$ fine :) Actually yes, because I am interested in the decay, for large $k$, so actually I need $|f(k)|\leq C/k^{1/2+\varepsilon}$ for large enough $k$. $\endgroup$ – Martingalo May 21 '15 at 19:59
  • $\begingroup$ Well, the first term will tend to dominate: $f(k) \approx e^{-k}$ as $k$ gets large. For $k = 6$, $f(k) = e^{-k}$ up to several digits. $\endgroup$ – Jair Taylor May 21 '15 at 20:11
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A slight refinement: $$ 0 < f(k) - e^{-k} = \sum_{k=2}^{\infty} e^{-kn^2} < \sum_{k=2}^{\infty} e^{-kn} = \frac{e^{-2k}}{1-e^{-k}}, $$ which is bounded by, say, $2e^{-2k}$ for $1-e^{-k}>1/2$, i.e. $k>\log{2}$. Hence, in the Poincaré-type asymptotics, $$ f(k) = e^{-k} + O(e^{-2k}). $$


Aside: The $(\pi/k)^{1/2}$ bound is useful when $k$ is positive and close to zero: it can be understood in terms of the Poisson summation formula: $$ \sum_{k = -\infty}^{\infty} e^{-\pi k n^2} = \frac{1}{\sqrt{k}} \sum_{k=-\infty}^{\infty} e^{-\pi n^2/k} $$

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There is no way to express in elementary functions. In fact, go to http://mathworld.wolfram.com/JacobiThetaFunctions.html and you will find $$ f(k)=\frac12(-1+\vartheta_3(0,e^{-k})). $$

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  • $\begingroup$ Aha! Interesting. Is there literature about the decay of such functions when $k$ is large? $\endgroup$ – Martingalo May 21 '15 at 20:04
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For $k\ge 1,$ $$\sum_{n=1}^{\infty}e^{-kn^2} \le \sum_{n=1}^{\infty}e^{-kn} = \frac{e^{-k}}{1-e^{-k}}\le \frac{1}{1-e^{-1}}\cdot e^{-k}.$$

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