It is written in a paper that I was reading that "by an application of the Poisson summation formula" we have
$\sum_{n \ne 0} |n|^{-1} e^{inx} = C \ln |x| + \phi(x)$
with some smooth function $\phi$. Unfortunately, I tried hard but I could not really see to which function we need to apply the Poisson formula.
