I haven't had analysis for a long time and I've forgotten plenty.
Could you please explain me and prove that the following converges:
$$ \sum_{n \geq 1}e^{-n^2t\pi}, t>0 c $$ and explain why the following function is holomorphic on $\text{Re}(s)>0$ $$ \int_{0}^{\infty}e^{-t}t^{s-1}dt $$