I'm having trouble following Ahlfors logic in his text concerning the proof that the $\wp$ function is convergent.
The proof comes down to whether the series $$\sum_{\omega\neq 0}\frac{1}{|\omega|^3}$$ is convergent.
He states that as $\omega_2/\omega_1$ is not real, there exists a $k>0$ such that $$\left | n_1\omega_1+n_2\omega_2\right |\geq k(|n_1|+|n_2|)\;\; \text{for all real pairs}\;\; (n_1,n_2)$$ Then concerning ourselves with just the integers there are $4n$ pairs $(n_1,n_2)$ with $|n_1|+|n_2|=n$ giving $$\sum_{\omega\neq 0}|\omega|^{-3}\leq 4k^{-3}\sum_1^{\infty}n^{-2}<\infty$$
Is there a proof for this? Or am I missing something trivial? Also is there a typo on the sum, shouldn't it be $n^{-3}$? If someone could point me in the right direction :)
I don't think its trivial. Here's a way to justify:
Define $f: \mathbb{R}^2\backslash\{0,0\} \to \mathbb{R}$ by $$f(n_1,n_2) = \frac{|n_1\omega_1+n_2\omega_2|}{|n_1|+|n_2|}$$ As $f(tn_1,tn_2)=f(n_1,n_2)$ for all $t \neq 0$, we can assume $n_1$, $n_2$ lie on the unit circle and restrict the domain of $f$. $f$ is continuous on a compact set so it attains its infimum, say $k$. As there are $4n$ pairs $(n_1,n_2)$ with $|n_1|+|n_2| = n$ we have $$\sum_{\omega \neq 0} \frac{1}{|\omega|^3} \leq \sum_{n=1}^\infty \frac{4n}{(kn)^3} = 4k^{-3} \sum_{n=1}^\infty \frac{1}{n^2}<\infty$$