I have to compute $$Z(t)=\sum_{n=1}^\infty e^{-\lambda_nt}$$ with $\lambda_n=\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)$. So $$\sum_{m,n=1}^\infty \exp\left(-\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)\right)t=\sum_{m=1}^\infty \exp\left(-\pi^2\left(\frac{m^2}{a^2}\right)t\right)\sum_{n=1}^\infty \exp\left(-\pi^2\left(\frac{n^2}{b^2}\right)t\right)$$
where $a$, $b$ are real constants and $t$ a real variable.
$$e^{-tn^2}\le\int_{n-1}^n e^{-tx^2}\le e^{-t(n-1)^2}$$ and $$e^{-t(n+1)^2}\le\int_{n}^{n+1} e^{-tx^2}\le e^{-t(n)^2}$$
so we obtain $$\int_{n}^{n+1} e^{-tx^2}\le e^{-t(n)^2} \le\int_{n-1}^n e^{-tx^2} \implies \int_{1}^{\infty} e^{-tx^2}\le \sum_{n \geq 1}e^{-t(n)^2} \le\int_{0}^{\infty} e^{-tx^2}$$ However, I cannot find explicitly the value $$\sum_{n=1}^\infty \exp\left(-\pi^2\left(\frac{n^2}{a^2}\right)t\right)$$ However, this does not give me an explicit solution on the convergence of $Z(t)$.
Could anyone could help me on how to compute $\sum_{n=1}^\infty \exp\left(-\pi^2\left(\frac{n^2}{a^2}\right)t\right)$?