3
votes
1answer
339 views

Calculating a summation of a $\theta$ function

Let $ \theta_z(t) = \sum \limits_{m,n\in\mathbb{z}}e^{-\pi Q_z(m,n)t}$ where $Q_z(m,n)=y^{-1}|mz+n|^2$. I need to prove that $\theta_z(t)=t^{-1}\theta_z(t^{-1})$. Now, looking that up I know that ...
5
votes
0answers
205 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...