This is a question in complex analysis that comes up in the treatment of large $N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed with $g$ the gauge coupling constant. In the physics literature one studies twopoint correlation functions as a function of fourmomentum $q$ and argues indirectly that the leading large $N$ behavior is given by a sum of poles, $\Pi(q)= \sum_i \frac{r_i}{q^2m_i^2}$. On the other hand one can compute directly at large $Q^2=q^2$ and finds the asymptotic behavior $\Pi(q) \sim c \log(Q^2)$ where $c$ is a computable constant. It is then claimed that these two results can be consistent only if the sum on $i$ is infinite. My question is whether the sum can be countable as is implicitly assumed in the physics literature. I realize this is probably not a research level question in mathematics, but standard texts on complex analysis that I am familiar with don't discuss such issues and I'm not sure where else to look.
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