How prove there exsit $C_{0}$ such $e^{-\pi^2 k^2\beta}-(\cos{(k\alpha\pi)}+\pi k\sin{(k\alpha\pi)})>C_{0}$ Question:

let $$\delta_{\alpha\beta}(k)=e^{-\pi^2 k^2\beta}-(\cos{(k\alpha\pi)}+\pi k\sin{(k\alpha\pi)})$$
if $\alpha\in Q$ and $\alpha>0,\beta>0,k\in N^{+}$,  show that
there exsits $C_{0}>0$, such that with sufficiently great $k$ the following inequality is valid:
$$|\delta_{\alpha\beta}(k)|\ge C_{0}>0$$

My idea: since
$$\delta_{\alpha\beta}(k)=e^{-\pi^2k^2\beta}-\sqrt{1+\pi^2k^2}\sin{(\pi k\alpha+\gamma_{k})},\gamma_{k}=\arcsin{\dfrac{1}{\sqrt{1+\pi^2k^2}}}$$
then I can't Continue  Thank you
 A: Since$$\lim_{k\rightarrow\infty}e^{-\pi^2 k^2\beta} = 0$$ for $\beta>0$, and $|\cos(k \alpha \pi)| \le 1$: for large values of $k \in \Bbb{N}$ the function $|\delta_{\alpha \beta}(k)|$ is dominated by the term $k \pi \sin(k \alpha \pi)$. Except of course when this term is zero. This happens exactly when
$$k \alpha \pi = 2 n \pi \Rightarrow k=2n/\alpha$$ for some integer $n$. These zeros will periodically land on an integer since $\alpha \in \Bbb{Q}$. At those times, however, we have $$|\delta_{\alpha \beta}(k)| = |e^{-\pi^2(2n/\alpha)^2\beta}-\cos(2n\pi)| =|e^{-4\pi^2 \frac{n^2}{\alpha^2}\beta}-1|=1-e^{-4\pi^2 \frac{n^2}{\alpha^2}\beta}$$
Pick any $0< C_0 <1$, we will have $0 < C_0 < |\delta_{\alpha \beta}(k)|$ whenever
$$1-C_0>e^{-4\pi^2 \frac{n^2}{\alpha^2}\beta} \Rightarrow n > \sqrt\frac{-\alpha^2\log(1-C_0)}{4\pi^2 \beta}$$ 
Or translating back into $k$ land when
$$ k > \frac{1}{\pi}\sqrt\frac{-\log(1-C_0)}{\beta}$$
Interestingly this does not depend upon $\alpha$ but of course it shouldn't; $\alpha$ just tells us where we need to be careful and $\beta$ determines the spread of the Gaussian which determines the value of $|\delta_{\alpha \beta}(k)|$  at the zeros of the sine function.
